Series $\sum\limits_{n=0}^{\infty}\frac{n+1}{2^n}$

Question

I want to check, whether $\sum\limits_{n=0}^{\infty}\frac{n+1}{2^n}$ converges or diverges.

I tried to use the Ratio test:

$|\frac{a_{n+1}}{a_n}|= \frac{n+2}{2^{n+1}} \frac{2^n}{n+1} = \frac{1}{2} \frac{n+1+1}{n+1} = \frac{1}{2} (1+ \frac{1}{n+1})$

$\lim\limits_{n \rightarrow \infty}{{(\frac{1}{2}(1+ \frac{1}{n+1})) = \frac{1}{2}}} \leq 1$

So $\sum\limits_{n=0}^{\infty}\frac{n+1}{2^n}$ converges.

Could somebody please check my solution?

It is almost perfect. You should have said that the limit is less than $1$, since if the limit is $1$ the ratio test is inconclusive. — André Nicolas, Dec 06 '13 at 20:21
Similar questions have been asked here before. You might have a look here, here or here; and also other questions linked to those questions. You will probably find more similar questions just by browsing frequent tab in the tags. sequences-and-series and summation. — Martin Sleziak, Dec 08 '13 at 11:56

score 2 · Answer 1 · answered Dec 06 '13 at 20:22

2

It looks fine to me. Perhaps you'd want to write it a bit neater but everything is there and the 'correct' test has been used. (the $\le$ doesn't matter so much since your limit is less than $1$ strictly. However, you should really have $<$).

answered Dec 06 '13 at 20:22

mathematics2x2life

13,322

score 2 · Answer 2 · answered Dec 06 '13 at 20:25

2

You can also break it down into a sum of two series $1/2^n $ and $n/2^n$ , each of which converges; first is a geometric series (with $r<1$), the second converges by , e.g., the zeta test; $n/2^n < 1/n^2$ for all $n>N$ ; $N$ a finite value.

answered Dec 06 '13 at 20:25

user99680

6,708

score 2 · Answer 3 · answered Dec 06 '13 at 20:40

The series converges, it's can be simply proved finding it's upper bound or as you've done it by the ratio test. But if you are interested in finding it's actual value here's a way to do it. From the sum of geometric series we have:

$$\sum_{n=0}^{\infty} r^n = \frac{1}{1-r}$$

Multiply both sides by $r$

$$\sum_{n=0}^{\infty} r^{n+1} = \frac{r}{1-r}$$

Differentiate with respect to $r$. Then we have:

$$\sum_{n=0}^{\infty} (n+1)r^n = \frac{1}{(r-1)^2}$$

Now substitute $r= \frac 12$. We have:

$$\sum_{n=0}^{\infty} \frac{n+1}{2^n} = \frac{1}{(-\frac 12)^2} = 4$$

This method is okay, but for those not familiar with this, one needs to either use formal power series (and formal differentiation) or check that we are differentiating a convergent power series. — user21820, Dec 26 '13 at 02:47

score 1 · Answer 4 · answered Dec 26 '13 at 03:12

$\sum_{n=0}^\infty (n+1) 2^{-n}$

$ = \sum_{n=0}^\infty \left( n 2^{-(n-1)} - (n+1) 2^{-n} + 2 \cdot 2^{-n} \right)$

$ = \left( 0 \cdot 2^{-(0-1)} - \lim_{n \to \infty} (n+1)2^{-n} \right) + 2 \sum_{n=0}^\infty { 2^{-n} }$

$ = 4$

The first step can be found be guesswork in simple cases, or be systematically doing "anti-difference by parts" in almost the same manner as "integration by parts", where the "by-parts" formula can be derived in exactly the same way from the "product rule" for the difference operator. Either of the forward or backwards difference operators work. This can easily handle things like $\sum_{n=0}^\infty n^3 3^{-n}$.

Series $\sum\limits_{n=0}^{\infty}\frac{n+1}{2^n}$

4 Answers4

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