# Is it true that #\sum_{n=1}^\infty a_n# converges (#a_n > 0# for every n) if and only if #\sum _{n=1}^\infty a_n^2 + a_n# converges?

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is it true that #\sum_{n=1}^\infty a_n# converges (#a_n > 0# for every n) if and only if #\sum _{n=1}^\infty a_n^2 + a_n# converges?

is it true that

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Andrea S.
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Feb 14, 2018

Suppose that:

with

so we can find a number

which implies:

Then by direct comparison also:

is convergent, and also:

is convergent, as it is the sum of two convergent series.

Conversely suppose that:

is convergent. Clearly as

and then by direct comparison also:

must be convergent.

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