Answer to Question #203264 in Real Analysis for Raj Kumar

Question #203264

Give an example of a series  ∑an such that  ∑an is not convergent but the sequence (an) converges to 0.


1
Expert's answer
2021-06-15T08:11:50-0400

Consider the harmonic series


i=1nan=i=1n1n\displaystyle\sum_{i=1}^na_n=\displaystyle\sum_{i=1}^n\dfrac{1}{n}

The sequence {an}\{a_n\} defined by an=1n,n1a_n=\dfrac{1}{n}, n\geq1 converges to 0.0.

But the series i=1n1n\displaystyle\sum_{i=1}^n\dfrac{1}{n} is divergent.



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