Determining Infinite Series Convergence from Partial Sum Formulas

Conditions

• You are given an explicit algebraic formula for the $n$-th partial sum, $S_n$.
• The limit is taken as $n \to \infty$.

Reasoning, step by step

1. Set up the limit of the partial sums: $S = \lim_{n \to \infty} S_n$.
2. Substitute the given formula for $S_n$ into the limit expression.
3. Evaluate the limit using algebraic techniques, such as comparing the degrees of the numerator and denominator.
4. Classify the series based on the result: a finite limit means convergence, while an infinite limit means divergence.

Example

Given $S_n = \frac{2n^3}{(n+1)(n+2)}$, the limit as $n \to \infty$ is $\infty$ because the numerator's degree (3) is greater than the denominator's degree (2). Therefore, the series diverges.

Common misconceptions

• Assuming the formula for $S_n$ is already the value of the infinite series.
• Believing that a series diverges only if its individual terms $a_n$ do not approach zero.

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Explore next

• How do you evaluate the limit at infinity of a rational function where the numerator has a higher degree than the denominator?
• What is the difference between the partial sum $S_n$ and the infinite series $S$?
• Why does the series diverge when the limit of the partial sums is infinite?