Find the limit of sequence $\frac{1}{2}+\frac{3}{2^{2}}+\frac{5}{2^{3}}+\cdots+\frac{2n-1}{2^{n}}$ without using of derivatives and etc.

Question

I need to find limit of sequence $$ \lim_{n \to \infty }\left(\frac{1}{2}+\frac{3}{2^{2}}+\frac{5}{2^{3}}+\cdots+\frac{2n-1}{2^{n}}\right) $$ I tried to solve it and stopped here $$ f(n+1) = \frac{1}{2}+\frac{3}{2^{2}}+\frac{5}{2^{3}}+\cdots+\frac{2n-1}{2^{n}}+\frac{2n+1}{2^{n+1}} $$ $$ 2f(n+1) = 1+\frac{3}{2}+\frac{5}{2^{2}}+\cdots+\frac{2n-1}{2^{n-1}}+\frac{2n+1}{2^{n}} $$ $$ 2f(n+1) -f(n) = 1+ \left(1 + \frac{1}{2}+\frac{1}{2^2}+\frac{1}{2^3}+\cdots+\frac{1}{2^{n-1}}\right) = 1 + g(n) $$ I can find the limit of $g$, but what to do with the other parts?

Answer 1

Let's assume that the limit exists. If we call the limit $s$ then

$s = \sum_1^{\infty}\frac{2n-1}{2^n}$

$\Rightarrow 2s = 1 + \sum_1^{\infty}\frac{2n+1}{2^n} = 1 + \sum_1^{\infty}\frac{2n-1}{2^n} + \sum_1^{\infty}\frac{2}{2^n}$

$\Rightarrow 2s = 1 + s + \sum_0^{\infty}\frac{1}{2^n}$

$\Rightarrow 2s = 1 + s + 2$

$\Rightarrow s=3$

So if the limit exists then it must be 3.

Now you just have to prove that the limit exists i.e. the series converges.

Answer 2

$$\lim_{n \to \infty }\left(\frac{1}{2}+\frac{3}{2^{2}}+\frac{5}{2^{3}}+...+\frac{2n-1}{2^{n}}\right)=\sum _1^{\infty} \frac {2n-1}{2^n}=2\sum _1^{\infty} \frac {n}{2^n} -\sum _1^{\infty} \frac {1}{2^n}=2\sum _1^{\infty} \frac {n}{2^n} -1$$

Note that

\begin{align} \sum _1^{\infty} \frac {n}{2^n}&=\left(\frac12+\frac14+\frac14+\frac18+\frac18+\frac18+\cdots\right)\\&=\left(\frac12+\frac14+\frac18 +\cdots\right) +\left(\frac14+\frac18+\frac1{16}+\cdots\right)+\left(\frac18+\frac1{16}+\frac1{32}+\cdots\right)+\cdots\\&=1+\frac12+\frac14+\frac18+\cdots=2 \end{align}

Thus we have $$\lim_{n \to \infty }\left(\frac{1}{2}+\frac{3}{2^{2}}+\frac{5}{2^{3}}+\cdots+\frac{2n-1}{2^{n}}\right)= 3$$

Answer 3

Using the identity $$ \frac{2k+1}{2^{k-1}}-\frac{2k+3}{2^k}=\frac{2k-1}{2^k}\tag1 $$ the sum can be written as a telescoping series $$ \begin{align} \sum_{k=1}^\infty\frac{2k-1}{2^k} &=\sum_{k=1}^\infty\left(\frac{2k+1}{2^{k-1}}-\frac{2k+3}{2^k}\right)\\ &=3\tag2 \end{align} $$

Answer 4

Use summation by parts. Given a sequence $f(n)$ define $(\Delta f)(n)=f(n+1)-f(n)$ and note that $$ \sum_{n=a}^b(\Delta f)(n)=f(b+1)-f(a). $$ Further given two sequences $f(n)$ and $g(n)$, note that $$ (\Delta fg)(n)=f(n+1)\Delta g(n)+g(n)\Delta f(n) $$ whence $$ \sum_{n=a}^bg(n)\Delta f(n)=[f(b+1)g(b+1)-g(a)f(a)]-\sum_{n=a}^bf(n+1)\Delta g(n)\tag{1} $$ Let $h(n)=-2/2^n$, and note that $\Delta h(n)=2^{-n}$. So by (1) $$ \sum_{n=1}^k\frac{2n-1}{2^n}=[(2k+1)(-2^{-k})-1(-1)]+2\sum_{k=1}^n2^{-k}.\tag{2} $$ Let $k\to \infty$ in (2) and use the formula for a geometric series to deduce that $$ \sum_{n=1}^\infty\frac{2n-1}{2^n}=1+2=3 $$