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

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?

• Hint: Alternating series – Anurag A Nov 3 '18 at 7:47
• Use Leibniz test – Corrêa Nov 3 '18 at 7:48
• Or, use the fact $\frac 1n\to0$ and group the term, use the comparison test. – Kemono Chen Nov 3 '18 at 7:52
• And another similar question. – rtybase Nov 3 '18 at 9:19

• 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 4 '18 at 7:53