Exercise 24Calculus › Single-variable calculus › Series › Convergence testsL1Added 2 Nov 2026

Equivalents, and a sign trap

#equivalents#comparison test#positive series

Statement

Determine the nature of
$\text{(a) }\sum_{n\ge1}\left(1-\cos\frac\pi n\right),\qquad\text{(b) }\sum_{n\ge1}\ln\left(1+\frac1n\right),\qquad\text{(c) }\sum_{n\ge2}\frac{(-1)^n+n}{n^2+1}.$

Hint

For positive terms, $u_n\sim v_n$ implies that $\sum u_n$ and $\sum v_n$ have the same nature. Use $1-\cos x\sim\frac{x^2}2$ and $\ln(1+x)\sim x$ at $0$ .

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Solution

All the considered series are positive.
(a) $1-\cos\frac\pi n\sim\frac{\pi^2}{2n^2}$ , and $\sum \frac{1}{n^2}$ converges: converges.

(b) $\ln\left(1+\frac1n\right)\sim\frac1n$ : diverges.

(c) The numerator $n+(-1)^n$ is positive for $n\ge2$ , and $\frac{n+(-1)^n}{n^2+1}\sim\frac1n$ : diverges.

The trap. In (c) the factor $(-1)^n$ looks like an alternating series, but it is only a small perturbation of the dominant term $n$ . The series is *not* alternating: all its terms are positive.

Equivalents, and a sign trap

Statement

Solution

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