How can $S_n=n$ for the series $S_n=\frac{1}{2}+\frac{3}{4}+\frac{7}{8}+\frac{15}{16}+\cdots$

Question

How can $S_n=n$ for the series $$S_n=\frac{1}{2}+\frac{3}{4}+\frac{7}{8}+\frac{15}{16}+\cdots$$

My try:

We can Re-write $S_n$ as

$$S_n=\frac{0+1}{2}+\frac{1+2}{2^2}+\frac{3+2^2}{2^3}+\frac{7+2^3}{2^4}+\cdots \frac{2^{n-1}-1+2^{n-1}}{2^n}$$

So we get

$$S_n=\frac{1}{2}+\frac{1}{2}+\cdots+\frac{1}{2}+\left(\frac{1}{2^2}+\frac{3}{2^3}+\frac{7}{2^4}+\cdots\right)$$

Hence

$$S_n=\frac{n}{2}+\frac{S_n}{2}$$

Hence

$$S_n=n$$

But the actual answer is entirely different. Whats wrong in my solution?

Answer 1

Your solution is not quite correct because $S_n$ has terms from $\frac{1}{2}$ to $\frac{2^n - 1}{2^n}$, but with your adjusted value, the terms on the right are missing the last value of $S_n$. Thus, the appropriate expression is$$\begin{equation}\begin{aligned} S_n & = \frac{n}{2} + \frac{S_n - \frac{2^{n} - 1}{2^{n}}}{2} \\ 2S_n & = n + S_n - \left(1 - \frac{1}{2^{n}}\right) \\ 2S_n & = n + S_n - 1 + \frac{1}{2^{n}} \\ S_n & = n - 1 + \frac{1}{2^{n}} \end{aligned}\end{equation}\tag{1}\label{eq1}$$

As an alternate solution method, note you have

$$\begin{equation}\begin{aligned} S_n & = \frac{2-1}{2} + \frac{4-1}{4} + \ldots + \frac{2^n-1}{2^n} \\ & = 1 - \frac{1}{2} + 1 - \frac{1}{4} + \ldots + 1 - \frac{1}{2^n} \\ & = (1 + 1 + \ldots + 1) - \left(\frac{1}{2} + \frac{1}{4} + \ldots + \frac{1}{2^n}\right) \\ & = n - \frac{1}{2}\left(\frac{1 - \frac{1}{2^n}}{1 - \frac{1}{2}}\right) \\ & = n - 1 + \frac{1}{2^n} \end{aligned}\end{equation}\tag{2}\label{eq2}$$

As you can see, this matches the corrected version of your solution attempt.

You can also check a few values to confirm, e.g., $S_1 = \frac{1}{2}$, $S_2 = 1 + \frac{1}{4} = \frac{5}{4} = \frac{1}{2} + \frac{3}{4}$, etc.

Answer 2

Given: $$S_n=\frac{1}{2}+\frac{3}{4}+\frac{7}{8}+\frac{15}{16}+\cdots+a_n,$$ as pointed out by John Omielan, you are missing the first (or last) term: $$S_n=\frac{\color{red}0+\color{blue}1}{2}+\frac{\color{red}1+\color{blue}2}{2^2}+\frac{\color{red}3+\color{blue}{2^2}}{2^3}+\frac{\color{red}7+\color{blue}{2^3}}{2^4}+\cdots +\underbrace{\frac{\color{red}{2^{n-1}-1}+\color{blue}{2^{n-1}}}{2^n}}_{a_n=\frac{2^n-1}{2^n}}=\\ \left(\frac{\color{red}0}2+\frac{\color{red}1}{2^2}+\frac{\color{red}3}{2^3}+\frac{\color{red}7}{2^4}+\cdots +\frac{\color{red}{2^{n-1}-1}}{2^n}\right)+ \left(\frac{\color{blue}1}2+\frac{\color{blue}2}{2^2}+\frac{\color{blue}{2^2}}{2^3}+\frac{\color{blue}{2^4}}{2^4}+\cdots +\frac{\color{blue}{2^{n-1}}}{2^n}\right)=\\ \frac12 \left(\color{red}0+\frac{\color{red}1}{2}+\frac{\color{red}3}{2^2}+\frac{\color{red}7}{2^3}+\cdots +\frac{\color{red}{2^{n-1}-1}}{2^{n-1}}\right)+ \left(\frac{\color{blue}1}2+\frac{\color{blue}1}{2}+\frac{\color{blue}{1}}{2}+\frac{\color{blue}{1}}{2}+\cdots +\frac{\color{blue}{1}}{2}\right)=\\ \frac12(0+S_{n-1})+\frac12n=\\ \frac12(S_n-a_n)+\frac12n \Rightarrow \\ S_n=n-a_n=n-\frac{2^{n-1}-1+2^{n-1}}{2^n}=n-\frac{2^n-1}{2^n}=n-\left(1-\frac{1}{2^n}\right)=n-1+\frac1{2^n}.$$ You can prove by mathematical induction: $S_{n}=n-1+\frac1{2^n}$. Indeed: $$S_n=\color{red}{S_{n-1}}+\color{blue}{a_n}=\color{red}{(n-1)-1+\frac1{2^{n-1}}}+\color{blue}{\frac{2^n-1}{2^n}}=\\ n-1+\frac{-2^n+2+2^n-1}{2^n}=n-1+\frac1{2^n}.$$