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Question Number 5948 by 123456 last updated on 06/Jun/16

is there a_{n} such that Σ a_{n} and Π a_{n}

converge ?

Answered by Yozzii last updated on 06/Jun/16

Y e s . L e t a_{n} = r^{n} w h e r e ∣ r ∣< 1 .

\Rightarrow \underset{n = 1}{\sum^{\infty}} r^{n} = \frac{r}{1 - r} a n d \underset{n = 1}{\prod^{\infty}} r^{n} = \lim_{N \to \infty} \underset{n = 1}{\prod^{N}} r^{n} = \lim_{N \to \infty} r^{0.5 N (N + 1)} = r^{\infty} = 0 .

S o, \exists a_{n} \in R s u c h t h a t Σ a_{n} a n d Π a_{n}

b o t h c o m v e r g e . M o r e i n t e r e s t i n g

e x a m p l e s c o u l d b e f o u n d .

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