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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?

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

Refer to alternating series test.

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  • $\begingroup$ 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? $\endgroup$ – TUC Nov 4 '18 at 7:53

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