3.2 Infinite Series and Convergence Tests

3.2 Infinite Series and Convergence Tests

Infinite series questions on the GRE Mathematics Subject Test evaluate theoretical criteria (absolute vs. conditional convergence, rearrangements) and rapid execution of convergence tests and closed-form summations.

Partial Sums and the Divergence Test

An infinite series $\sum_{n=1}^\infty a_n$ converges to sum $S \in \mathbb{R}$ if its partial sums $S_N = \sum_{n=1}^N a_n$ satisfy $\lim_{N \to \infty} S_N = S$. Otherwise, it diverges.

The Divergence Test ($n$-th Term Test)

If $\lim_{n \to \infty} a_n \ne 0$, or if the limit fails to exist, then $\sum a_n$ diverges.

• Warning: $\lim a_n = 0$ does not imply convergence. The harmonic series $\sum 1/n$ has $1/n \to 0$, yet diverges.

Geometric and Telescoping Series

When asked for the exact value of a series, look for geometric or telescoping structure:

• Geometric Series: $\sum_{n=0}^\infty a r^n = \frac{a}{1 - r}$ for $|r| < 1$. Diverges for $|r| \ge 1$.
• Telescoping Series: Partial sums collapse via cancellation: $S_N = \sum_{n=1}^N (b_n - b_{n+1}) = b_1 - b_{N+1} \implies \sum_{n=1}^\infty (b_n - b_{n+1}) = b_1 - \lim_{N \to \infty} b_{N+1}$ Use partial fractions on rational expressions to expose cancellations.

Benchmark Series: $p$-Series and Cauchy Condensation

• $p$-Series: $\sum_{n=1}^\infty \frac{1}{n^p}$ converges if and only if $p > 1$.
• Logarithmic $p$-Series: $\sum_{n=2}^\infty \frac{1}{n (\ln n)^p}$ converges if and only if $p > 1$ (diverges for $p \le 1$).
• Cauchy Condensation: For non-negative, non-increasing $a_n$, $\sum a_n$ converges $\iff \sum 2^k a_{2^k}$ converges.

Comparison Tests for Non-Negative Series

Let $a_n, b_n \ge 0$:

1. Direct Comparison Test (DCT):
   • If $a_n \le b_n$ and $\sum b_n$ converges, then $\sum a_n$ converges.
   • If $a_n \ge b_n$ and $\sum b_n$ diverges, then $\sum a_n$ diverges.
2. Limit Comparison Test (LCT): Compute $L = \lim_{n \to \infty} \frac{a_n}{b_n}$.
   • If $0 < L < \infty$: Both series converge or both diverge.
   • If $L = 0$ and $\sum b_n$ converges: $\sum a_n$ converges.
   • If $L = \infty$ and $\sum b_n$ diverges: $\sum a_n$ diverges. GRE Strategy: For algebraic/rational expressions, drop lower-order terms to choose $b_n = n^{p_{\text{num}} - p_{\text{den}}}$.

The Integral Test and Remainder Bounds

If $f(x)$ is continuous, positive, and decreasing on $[1, \infty)$ with $f(n) = a_n$, then $\sum a_n$ and $\int_1^\infty f(x),dx$ either both converge or both diverge.

• Remainder Estimate: The truncation remainder $R_N = S - S_N$ satisfies: $\int_{N+1}^\infty f(x)\,dx \le R_N \le \int_N^\infty f(x)\,dx$

Alternating Series Test (Leibniz) and Error Bounds

An alternating series $\sum_{n=1}^\infty (-1)^{n-1} b_n$ with $b_n > 0$ converges if:

1. $b_{n+1} \le b_n$ for all $n \ge N$, and
2. $\lim_{n \to \infty} b_n = 0$.

Alternating Series Estimation Theorem

The error in truncating at $N$ terms is bounded by the first neglected term: $|R_N| = |S - S_N| \le b_{N+1}$. Furthermore, $S$ lies between any two consecutive partial sums $S_N$ and $S_{N+1}$.

Absolute vs. Conditional Convergence

• $\sum a_n$ is absolutely convergent if $\sum |a_n|$ converges.
• $\sum a_n$ is conditionally convergent if $\sum a_n$ converges, but $\sum |a_n|$ diverges.
• Theorem: Absolute convergence implies convergence ($\sum |a_n| < \infty \implies \sum a_n \text{ converges}$).
• Riemann Rearrangement Theorem: Absolutely convergent series converge to the same sum under all rearrangements; conditionally convergent series can be rearranged to sum to any real $M \in [-\infty, \infty]$.

Ratio and Root Tests

Let $L = \lim_{n \to \infty} \left|\frac{a_{n+1}}{a_n}\right|$ (Ratio) or $L = \limsup_{n \to \infty} |a_n|^{1/n}$ (Root):

• If $L < 1$: Series converges absolutely.
• If $L > 1$: Series diverges.
• If $L = 1$: Inconclusive (fails on all rational and $p$-series terms). Note: The Root Test is strictly stronger than or equivalent to the Ratio Test.

Summary Comparison Table: Convergence Tests

Worked Problem: Evaluating a Telescoping Series

Problem: Evaluate $\sum_{n=1}^\infty \frac{2}{n^2 + 2n}$.

Solution:

1. Apply partial fractions: $\frac{2}{n(n+2)} = \frac{1}{n} - \frac{1}{n+2}$.
2. Write the $N$-th partial sum: $S_N = \sum_{n=1}^N \left(\frac{1}{n} - \frac{1}{n+2}\right) = \left(1 - \frac{1}{3}\right) + \left(\frac{1}{2} - \frac{1}{4}\right) + \cdots + \left(\frac{1}{N-1} - \frac{1}{N+1}\right) + \left(\frac{1}{N} - \frac{1}{N+2}\right)$
3. Internal terms cancel, leaving $S_N = 1 + \frac{1}{2} - \frac{1}{N+1} - \frac{1}{N+2} = \frac{3}{2} - \frac{1}{N+1} - \frac{1}{N+2}$.
4. Taking $N \to \infty$: $\lim_{N \to \infty} S_N = \frac{3}{2}$.

GRE Exam Traps & Fast Test-Taking Strategies

• Trap 1: Applying Ratio Test to rational series. Ratios of polynomials always yield $L = 1$. Use Limit Comparison with a $p$-series immediately.
• Trap 2: Ignoring monotonicity in Leibniz's test. Alternating signs with $a_n \to 0$ is not enough; magnitudes must decrease monotonically. $\sum \frac{(-1)^n}{\sqrt{n} + (-1)^n}$ diverges despite $a_n \to 0$.
• Trap 3: Using alternating error bounds on positive series. The bound $|R_N| \le a_{N+1}$ requires an alternating series. For positive series, use the Integral Test remainder bound.