Prove a series $\sum_{n=1} ^\infty \frac{(-1)^n}{n}$ is convergent

Question

I was asked to prove that an infinite series $$\sum_{n=1} ^\infty \frac{(-1)^n}{n}$$ is a convergent series.

I tried using ratio test but the limit results in 1 which is inconclusive.

I am stuck at this point.

Can you give me some hint on how to approach this question?

Thank you

edit : My professor only taught ratio test, root test and comparison test where if $|b_i| \leq a_i$ for all i = 1, 2, ... and $\sum a_i$ converges then the sum of $b_i$ converges absolutely. Is there any way other than alternating series test to prove this problem?

Or, use the fact $\frac 1n\to0$ and group the term, use the comparison test. — Kemono Chen, Nov 03 '18 at 07:52
@J.G. What a pity you deleted your answer! I think it was the most appropriate, requiring few knowledge. — Anne Bauval, Feb 08 '24 at 03:32

score 1 · Answer 1 · answered Nov 03 '18 at 07:48

1

HINT

Refer to alternating series test.

answered Nov 03 '18 at 07:48

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My professor only taught ratio test, root test and comparison test where if |$b_i$| $\leq$ $a_i$ for all i = 1, 2, ... and $\sum a_i$ converges then the sum of $b_i$ converges absolutely. Is there any way other than alternating series test to prove this problem? – TUC Nov 04 '18 at 07:53

Prove a series $\sum_{n=1} ^\infty \frac{(-1)^n}{n}$ is convergent

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