# 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 $\setminus {\sum}_{n = 1}^{\setminus} \infty {a}_{n}$ converges (${a}_{n} > 0$ for every n) if and only if $\setminus {\sum}_{n = 1}^{\setminus} \infty {a}_{n}^{2} + {a}_{n}$ converges?

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Feb 14, 2018

Suppose that:

${\sum}_{n = 1}^{\infty} {a}_{n}$

with ${a}_{n} > 0$ is convergent. Then necessarily:

${\lim}_{n \to \infty} {a}_{n} = 0$

so we can find a number $N$ such that:

$n > N \implies 0 < {a}_{n} < 1$

which implies:

$n > N \implies 0 < {a}_{n}^{2} < {a}_{n}$

Then by direct comparison also:

${\sum}_{n = 1}^{\infty} {a}_{n}^{2}$

is convergent, and also:

${\sum}_{n = 1}^{\infty} {a}_{n}^{2} + {a}_{n}$

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

Conversely suppose that:

${\sum}_{n = 1}^{\infty} {a}_{n}^{2} + {a}_{n}$

is convergent. Clearly as ${a}_{n}^{2} > 0$ we have:

$0 < {a}_{n} < {a}_{n} + {a}_{n}^{2}$

and then by direct comparison also:

${\sum}_{n = 1}^{\infty} {a}_{n}$

must be convergent.

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