Convergence of a series knowing information of convergence of the summands

Asked Jul 26 '13 at 02:25

Active Jul 26 '13 at 02:50

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I have the series $F(n)=\sum_{j=0}^n = f(n-j)g(j)$ and I have that

1- As $m$ increases, $f(m)$ approaches $L$ (the limit point) 2- As $j$ increases, $g(j)$ decreases EXPONENTIALLY to $0$

Question: Can I say that $\lim_{n\to \infty} F(n) = \sum_{j=0}^{\infty} L g(j)$

If not, does the series converge?

Thank you

edited Jul 26 '13 at 02:42

asked Jul 26 '13 at 02:25

user87757

1

You need to assume the series of $g(i)$ converges, first. – Pedro Jul 26 '13 at 02:32
I mentioned that $g(j) \ to 0$ as $j \to 0$ – user87757 Jul 26 '13 at 02:38
That is not enough. The canonical example is $$g(j)=\frac{1}{j+1}$$ – Pedro Jul 26 '13 at 02:41
my bad! $g(j)$ decreases exponentially – user87757 Jul 26 '13 at 02:42
You can just assume it converges, you lose nothing by that! =) – Pedro Jul 26 '13 at 02:43
Of course, my point of interest is to see the convergence of that. Read the title O_O – user87757 Jul 26 '13 at 03:01
If the $g(j)$ are positive, and $\sum g(j)$ converges, everything will be OK. The (currently) deleted answer by Peter Tamaroff, with a little detail filled in, will do it. – André Nicolas Jul 26 '13 at 04:34

0 Answers0