Difference Between Partial Sum $S_n$ and Infinite Series $S$

Conditions

• $S_n$ denotes the sum of the first $n$ terms.
• $S$ denotes the infinite series $\sum_{n=1}^{\infty} a_n$.
• The limit is taken as $n \to \infty$.

Reasoning, step by step

1. Identify $S_n$ as the cumulative sum of a finite number of terms.
2. Identify $S$ as the theoretical sum of infinitely many terms.
3. Relate the two by stating that $S = \lim_{n \to \infty} S_n$.
4. Conclude that $S_n$ is the stepping stone to finding $S$, not $S$ itself.

Example

The video emphasizes that $S_n$ gives the sum of the first $n$ terms, while the infinite series is obtained only after taking $\lim_{n\to\infty} S_n$. One might mistakenly think the formula for $S_n$ is already the value of the infinite series.

Common misconceptions

• Confusing the partial sum formula with the infinite series itself.
• Assuming that evaluating $S_n$ for a large $n$ gives the exact value of $S$.

Watch the explanation

Explore next

• How do you determine if an infinite series converges or diverges when given a formula for its partial sums?
• Why does the series diverge when the limit of the partial sums is infinite?
• What is the definition of an infinite series?