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 #\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 #\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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Feb 14, 2018

Suppose that:

#sum_(n=1)^oo a_n#

with #a_n > 0# is convergent. Then necessarily:

#lim_(n->oo) a_n = 0#

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

#n > N => 0 < a_n < 1#

which implies:

#n > N => 0< a_n^2 < a_n#

Then by direct comparison also:

#sum_(n=1)^oo a_n^2#

is convergent, and also:

#sum_(n=1)^oo a_n^2+a_n#

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

Conversely suppose that:

#sum_(n=1)^oo 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)^oo a_n#

must be convergent.

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