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I'm confused on how Convergence does not mean the same thing as the sum of a series.

I was asked to find the sum of $\sum_{0}^{\infty}\frac{n+1}{2^n}$. I found that it converged to $\frac{1}{2}$, but that its sum was $4$. I thought convergence described the end-behavior of a function and that $\frac{1}{2}$ would be the number that it would tend to.

Can someone explain the difference in Convergence and the infinite sum of a series?

Martin Sleziak
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user113027
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  • Perhaps you could explain what you mean when you write: I found that it converged to $\frac12$. If you mean the $n$-th term, the fraction $\frac{n+1}{2^n}$ converges to zero. – Martin Sleziak Dec 08 '13 at 11:58
  • BTW you can find here several questions about very similar series, for example http://math.stackexchange.com/questions/595981/series-sum-limits-n-0-infty-fracn12n or http://math.stackexchange.com/questions/30732/how-can-i-evaluate-sum-n-0-infty-n1xn (And other questions linked there.) – Martin Sleziak Dec 08 '13 at 11:59

3 Answers3

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Maybe you're referring to the fact that a series is a sequence of partial sums? I mean $S_n:= \Sigma_{i=1}^n a_i $ . Then the sum converges if the sequence $S_n$ of partial sums converges. On the other hand, a sequence $a_n$ converges to $a$, if it gets indefinitely-close to $a$ , in a $\delta - \epsilon$ sense.

user99680
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Convergence just means the sequence of partial sums has a finite limit. What that limit is, is the sum.

It seems to me that you are confusing the idea of what the terms $a_n$ are converging to and what the sum of those terms is converging to.

For example, $\displaystyle\sum_{n=0}^\infty (1/2)^n$ has $a_n=(1/2)^n\to 0$ as $n\to\infty$, but the sum of the series is $2$ since $\displaystyle s_k=\sum_{n=0}^k (1/2)^n\to 2$ as $k\to\infty$.

JohnD
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  • So for your example, $\frac{1}{2}^n$ converges to $0$, but series sums to $2$? So what significance is there in finding the number to which a series converges to? – user113027 Dec 06 '13 at 21:10
  • If $a_n\not\to 0$, then $\sum_{n=1}^\infty a_n$ diverges. Note, however, that just because $a_n\to 0$, that $\sum_n a_n$ doesn't necessarily converge (e.g. harmonic series). – JohnD Dec 06 '13 at 23:12
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Convergence means that the sum approaches a single, finite value.

The sum of the series is the finite value toward which it converges.

(That sum doesn't converge to $1/2$, by the way. The first term is $1$, and all of the terms are positive.)

John
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