Infinite series as limit of partial sums | Series | AP Calculus BC | Khan Academy

1. Formula
   Observation

   The symbol $S$ is written in yellow at the top left and used to denote the infinite series.

2. Audio
   Observation

   The speaker says, "Let's say that we have an infinite series S."

S

The sum of the infinite series.

Real numbers or extended real numbers (infinity).

1. Formula
   Observation

   The index $n$ appears in the summation limits $n=1$ to $\infty$ and in the partial sum formula $S_n$.

2. Audio
   Observation

   The speaker refers to "the sum from n equals 1 to infinity" and "the sum of the first n terms".

n

The index of summation and the number of terms in the partial sum.

Positive integers ($n \ge 1$).

1. Formula
   Observation

   The term $a_n$ is written inside the summation $\sum_{n=1}^{\infty} a_n$.

2. Audio
   Observation

   The speaker says, "of a sub n".

$a_n$

The general term of the infinite series.

Real numbers.

1. Formula
   Observation

   The symbol $S_n$ is written in purple and defined by the formula $S_n = \frac{2n^3}{(n+1)(n+2)}$.

2. Audio
   Observation

   The speaker says, "we have a formula for the partial sums of S... S sub n is equal to..."

$S_n$

The $n$-th partial sum of the series $S$, representing the sum of the first $n$ terms.

Real numbers.

1. Formula
   Observation

   $S = \sum_{n=1}^{\infty} a_n = a_1 + a_2 + \cdots$

S

the infinite series under discussion

real-valued series

1. Formula
   Observation

   $a_n$ appears in $S = \sum_{n=1}^{\infty} a_n$

$a_n$

the nth term of the series

sequence indexed by positive integers

1. Formula
   Observation

   $S_n = \frac{2n^3}{(n+1)(n+2)}$ and later $S = \lim_{n\to\infty} S_n$

$S_n$

the nth partial sum of the series

positive integers n

1. Formula
   Observation

   n is the index in $\sum_{n=1}^{\infty} a_n$, $S_n$, and $\lim_{n\to\infty}$

n

positive integer index used for terms, partial sums, and the limiting process

$n \ge 1$

1. Formula
   Observation

   The equation $S = \sum_{n=1}^{\infty} a_n = a_1 + a_2 + \dots$ is written on the screen.

2. Audio
   Observation

   The speaker explains that the series goes on and on forever.

An infinite series is the sum of an infinite sequence of terms. It is denoted by $S$ and can be written using summation notation or expanded as a sum of individual terms.

1. The index $n$ starts at 1 and approaches infinity.

1. Formula
   Observation

   The equation $S_n = \frac{2n^3}{(n+1)(n+2)}$ is written in purple below the series definition.

2. Audio
   Observation

   The speaker states this is the formula for the partial sums of $S$.

The problem provides a specific algebraic formula for the $n$-th partial sum $S_n$, which represents the sum of the first $n$ terms of the series.

1. $n$ is a positive integer.

1. Definition of an Infinite Series

1. Formula
   Observation

   The equation $S = \lim_{n \to \infty} S_n$ is written in yellow.

2. Audio
   Observation

   The speaker explains that the infinite series $S$ is just the limit as $n$ approaches infinity of the partial sums.

An infinite series converges to a finite value $S$ if the sequence of its partial sums $S_n$ approaches a limit as $n$ goes to infinity. If the limit does not exist or is infinite, the series diverges.

1. The limit must exist and be finite for convergence.

1. Definition of an Infinite Series
2. Formula for Partial Sums

1. Formula
   Observation

   $S = \lim_{n\to\infty} S_n$

2. Audio
   Observation

   The speaker says the infinite series can be viewed as the limit as n approaches infinity of the partial sum $S_n$.

The video treats the value of an infinite series S as the limit of its sequence of partial sums $S_n$. If that limit exists as a finite number, the series converges to it; if the limit grows without bound, the series diverges.

1. $S_n$ denotes the nth partial sum

2. the limit is taken as $n \to \infty$

1. Formula
   Observation

   $S_n = \frac{2n^3}{(n+1)(n+2)}$

For this example, the nth partial sum is explicitly given by a rational expression in n.

1. n is a positive integer

1. Infinite series as the limit of its partial sums

1. Audio
   Observation

   we can divide the numerator and the denominator by n squared

2. Formula
   Observation

   $\lim_{n\to\infty} \frac{2n^3}{n^2+3n+2} = \lim_{n\to\infty} \frac{2n}{1+\frac{3}{n}+\frac{2}{n^2}}$

To evaluate the limit of a rational expression as n goes to infinity, the video divides both numerator and denominator by the highest power of n appearing in the denominator, here $n^2$, so that lower-order terms become fractions with n in the denominator.

1. the expression is a quotient of polynomials in n

2. $n \to \infty$

1. Given formula for the nth partial sum

1. Audio
   Observation

   since the limit of the partial sums goes to infinity, that means that this infinite series is not going to be a finite value. It's just going to diverge.

2. Audio
   Observation

   In order for it to have converged, this limit should have been some finite value.

An infinite series converges only if the limit of its partial sums is a finite value; if that limit is infinite, the series diverges.

1. S is represented as $\lim_{n\to\infty} S_n$

for the series under discussion

1. Formula
   Observation

   The speaker writes $\lim_{n \to \infty} \frac{2n^3}{(n+1)(n+2)}$ and then expands the denominator to $n^2 + 3n + 2$.

2. Audio
   Observation

   The speaker compares the degrees of the numerator (3) and denominator (2) to conclude the limit is infinity.

1. Expression

   $$\lim_{n \to \infty} S_n = \lim_{n \to \infty} \frac{2n^3}{(n+1)(n+2)}$$
   Explanation

   Substitute the given formula for the partial sum $S_n$ into the limit definition of the series sum.

   Justification

   Definition of series convergence and given formula for $S_n$.

   Shown in the video
2. Expression

   $$\lim_{n \to \infty} \frac{2n^3}{n^2 + 3n + 2}$$
   Explanation

   Expand the denominator $(n+1)(n+2)$ to get $n^2 + 3n + 2$.

   Justification

   Algebraic expansion.

   Shown in the video
3. Expression

   $$\text{Degree of numerator} = 3, \text{ Degree of denominator} = 2$$
   Explanation

   Identify the highest power of $n$ in the numerator and the denominator.

   Justification

   Polynomial degree analysis.

   Shown in the video
4. Expression

   $$\lim_{n \to \infty} \frac{2n^3}{n^2 + 3n + 2} = \infty$$
   Explanation

   Since the degree of the numerator is greater than the degree of the denominator, the rational function grows without bound as $n$ approaches infinity.

   Justification

   Properties of limits of rational functions at infinity.

   Shown in the video

The limit of the partial sums is infinity, so the series diverges.

1. Formula
   Observation

   $S = \lim_{n\to\infty} S_n = \lim_{n\to\infty} \frac{2n^3}{(n+1)(n+2)} = \lim_{n\to\infty} \frac{2n^3}{n^2+3n+2}$

2. Formula
   Observation

   = $\lim_{n\to\infty} \frac{2n}{1+\frac{3}{n}+\frac{2}{n^2}} = \infty$

3. Audio
   Observation

   this thing as n approaches infinity ... is going to go towards infinity ... the denominator is going to go towards 1 ... the limit is going to go to infinity

1. Expression

   $$S = \lim_{n\to\infty} S_n$$
   Explanation

   Rewrite the infinite series as the limit of its partial sums.

   Justification

   Definition stated on screen and in narration.

   Shown in the video
2. Expression

   $$= \lim_{n\to\infty} \frac{2n^3}{(n+1)(n+2)}$$
   Explanation

   Substitute the given formula for $S_n$.

   Justification

   Given formula shown on screen.

   Shown in the video
3. Expression

   $$= \lim_{n\to\infty} \frac{2n^3}{n^2+3n+2}$$
   Explanation

   Expand the denominator ($n+1$)($n+2$).

   Justification

   Algebraic expansion visible in the written work.

   Shown in the video
4. Expression

   $$= \lim_{n\to\infty} \frac{2n}{1+\frac{3}{n}+\frac{2}{n^2}}$$
   Explanation

   Divide numerator and denominator by $n^2$.

   Justification

   Stated aloud and shown in the rewritten fraction.

   Shown in the video
5. Expression

   $$= \infty$$
   Explanation

   As $n \to \infty$, the numerator $2n \to \infty$ while $\frac{3}{n} \to 0$ and $\frac{2}{n^2} \to 0$, so the denominator tends to 1.

   Justification

   Limit behavior explained verbally and annotated on screen.

   Shown in the video

The limit of the partial sums is infinite, so the series diverges.

1. Audio
   Observation

   The speaker poses the question: "Does this series converge or diverge?"

2. Formula
   Observation

   The entire derivation on the screen solves this specific problem.

Given an infinite series $S$ with partial sums $S_n = \frac{2n^3}{(n+1)(n+2)}$, determine if the series converges or diverges.

1. $S = \sum_{n=1}^{\infty} a_n$

2. $S_n = \frac{2n^3}{(n+1)(n+2)}$

Find $\lim_{n \to \infty} S_n$ to determine convergence.

1. Expression

   $$S = \lim_{n \to \infty} S_n$$
   Explanation

   Set up the limit of the partial sums.

   Justification

   Definition of the sum of an infinite series.

   Shown in the video
2. Expression

   $$\lim_{n \to \infty} \frac{2n^3}{(n+1)(n+2)}$$
   Explanation

   Substitute the expression for $S_n$.

   Justification

   Given formula.

   Shown in the video
3. Expression

   $$\lim_{n \to \infty} \frac{2n^3}{n^2 + 3n + 2}$$
   Explanation

   Expand the denominator.

   Justification

   Algebra.

   Shown in the video
4. Expression

   $$\infty$$
   Explanation

   Evaluate the limit based on the degrees of the polynomials.

   Justification

   Numerator degree (3) > Denominator degree (2).

   Shown in the video

The series diverges because the limit of its partial sums is infinity.

The visual comparison of polynomial degrees confirms the unbounded growth.

1. Formula
   Observation

   $S = \sum_{n=1}^{\infty} a_n = a_1 + a_2 + \cdots$, $S_n = \frac{2n^3}{(n+1)(n+2)}$

2. Audio
   Observation

   this infinite series is not going to be a finite value. It's just going to diverge.

Given $S = \sum_{n=1}^{\infty} a_n$ and $S_n = \frac{2n^3}{(n+1)(n+2)}$, determine whether the infinite series converges or diverges.

1. $S = \sum_{n=1}^{\infty} a_n = a_1 + a_2 + \cdots$

2. $S_n = \frac{2n^3}{(n+1)(n+2)}$

Evaluate $\lim_{n\to\infty} S_n$ and infer convergence or divergence.

1. Expression

   $$S = \lim_{n\to\infty} S_n$$
   Explanation

   Use the definition of an infinite series as the limit of its partial sums.

   Justification

   Shown on screen and restated in narration.

   Shown in the video
2. Expression

   $$\lim_{n\to\infty} \frac{2n^3}{(n+1)(n+2)}$$
   Explanation

   Substitute the explicit formula for $S_n$.

   Justification

   Given in the problem statement.

   Shown in the video
3. Expression

   $$\lim_{n\to\infty} \frac{2n^3}{n^2+3n+2}$$
   Explanation

   Expand the denominator.

   Justification

   Algebraic simplification shown in the worked solution.

   Shown in the video
4. Expression

   $$\lim_{n\to\infty} \frac{2n}{1+\frac{3}{n}+\frac{2}{n^2}}$$
   Explanation

   Divide numerator and denominator by $n^2$ to expose the limiting behavior.

   Justification

   Method explained aloud and written on screen.

   Shown in the video
5. Expression

   $$\infty$$
   Explanation

   The numerator grows without bound while the denominator approaches 1.

   Justification

   Limit analysis stated in the audio and annotated visually.

   Shown in the video

The series diverges.

The conclusion matches the final on-screen annotation DIVERGE and the spoken explanation that convergence would require a finite limit.

1. Animation
   Observation

   The speaker writes $S_1, S_2, S_3, \dots$ in pink on the right side of the screen.

2. Audio
   Observation

   The speaker describes this as a sequence of partial sums.

1. Text $S_1, S_2, S_3, \dots$

1. The sequence is written out term by term.

1. The concept that these are discrete values approaching a limit.

This visual aid helps conceptualize the limit process by showing the individual partial sums as a sequence before taking the limit.

1. Diagram
   Observation

   Black digital whiteboard with yellow, purple, pink, and green handwritten mathematics arranged in rows.

1. yellow series definition at top left

2. purple partial-sum formula below it

3. pink sequence notation $S_1$, $S_2$, $S_3$, $\ldots$ at upper right

4. yellow limit chain across the middle

5. green box and arrow near the end

1. new algebraic lines are written beneath the initial setup

2. annotations $\infty$, 0, 0, and 1 are added around the simplified limit

3. a green box is drawn around S and the word DIVERGE is written with an arrow

1. the original definitions and formulas remain visible throughout

The visual progression mirrors the reasoning chain from definition to substitution, simplification, limit evaluation, and final classification.

1. Diagram
   Observation

   Annotations appear above and beside parts of $\frac{2n}{1+\frac{3}{n}+\frac{2}{n^2}}$ indicating $\infty$ for 2n, 0 for $\frac{3}{n}$, 0 for $\frac{2}{n^2}$, and 1 for the denominator.

1. numerator 2n

2. terms $\frac{3}{n}$ and $\frac{2}{n^2}$

3. denominator $1+\frac{3}{n}+\frac{2}{n^2}$

1. arrows and labels mark each part's limit as $n \to \infty$

1. the algebraic form of the fraction stays the same while its limiting behavior is highlighted

The annotations show why the whole fraction tends to infinity: the numerator diverges while the denominator tends to 1.

1. Diagram
   Observation

   A green box is drawn around S and an arrow points to the handwritten word DIVERGE.

1. boxed S

2. arrow

3. word DIVERGE

1. the conclusion is emphasized graphically after the limit computation

1. the earlier derivation remains visible above

The board marks the final result of the example: the infinite series does not converge to a finite value.

1. Audio
   Observation

   Always says look, infinite series, we had a formula for the partial sum of the first n terms, and then we said oh look, the series itself, the infinite series, you could view it as the limit ... of the partial sum $S_n$.

One might think the formula for $S_n$ is already the value of the infinite series.

The video emphasizes that $S_n$ gives the sum of the first n terms, while the infinite series is obtained only after taking $\lim_{n\to\infty} S_n$.

1. Audio
   Observation

   the denominator is going to go towards 1

2. Formula
   Observation

   $\lim_{n\to\infty} \frac{2n}{1+\frac{3}{n}+\frac{2}{n^2}} = \infty$

1. This misconception is inferred from the explanatory emphasis rather than explicitly named by the speaker.

After seeing a quadratic denominator, one may expect the limit to be 0.

Because the numerator is cubic, division by $n^2$ leaves a linear numerator 2n, so the fraction still grows without bound even though the denominator approaches 1.

1. Audio
   Observation

   The speaker explicitly links the infinite series $S$ to the limit of $S_n$.

The convergence of an infinite series is defined by applying the concept of limits to its sequence of partial sums.

1. Formula
   Observation

   $S = \lim_{n\to\infty} S_n$ is written directly above the substituted expression for $S_n$

The example applies the definition of an infinite series as the limit of its partial sums.

1. Formula
   Observation

   The given $S_n$ is transformed into a simplified limit expression by dividing by $n^2$

The specific rational formula for $S_n$ is analyzed using the method of dividing numerator and denominator by the highest power of n.

1. Audio
   Observation

   we can divide the numerator and the denominator by n squared

2. Formula
   Observation

   the rewritten limit follows immediately in the derivation

The derivation depends on the algebraic rewriting method to expose the limiting behavior of each term.

1. Audio
   Observation

   since the limit of the partial sums goes to infinity ... it's just going to diverge

The computed infinite limit is used to justify the claim that the series diverges.

1. Audio
   Observation

   The speaker asks, "Does this series converge or diverge?"

1. Convergence of an Infinite Series
2. Formula for Partial Sums

1. Audio
   Observation

   The speaker compares the degrees of the numerator and denominator to evaluate the limit.

1. Evaluating the Limit of Partial Sums

1. Formula
   Observation

   $S = \lim_{n\to\infty} S_n$

1. Infinite series as the limit of its partial sums

1. Audio
   Observation

   divide the numerator and the denominator by n squared

2. Formula
   Observation

   $\lim_{n\to\infty} \frac{2n}{1+\frac{3}{n}+\frac{2}{n^2}} = \infty$

1. Dividing numerator and denominator by the highest power of n
2. Evaluation of the limit of the partial sums

1. Audio
   Observation

   this infinite series is not going to be a finite value. It's just going to diverge.

1. Convergence criterion stated in the clip
2. Example: deciding convergence from a given partial-sum formula

1. Audio
   Observation

   we had a formula for the partial sum of the first n terms ... the infinite series, you could view it as the limit ... of the partial sum $S_n$

1. Infinite series as the limit of its partial sums
2. Confusing the partial sum formula with the infinite series itself