How do I sum this summation?

Question

How do I find the sum:

$$\sum_{n=0}^{1947}{\frac{1}{2^n+\sqrt{2^{1947}}}}$$

The method of telescoping surely won't work for this, and I tried rationalizing each terms to try and cancel consecutive terms but it didnt work for me. How do I go about approaching this?

Answer 1

We have

$$\sum_{n=0}^{1947} \frac{1}{ 2^n+\sqrt{2^{1947}} }=\frac1{\sqrt{2^{1947}} }\sum_{n=0}^{1947} \frac{1}{2^{n-\frac{1947}2}+1}=\frac1{\sqrt{2^{1947}}}\sum_{n=0}^{1947}a_n$$

then

$$a_0+a_{1947}=\frac{1}{2^{-\frac{1947}2}+1}+\frac{1}{2^{\frac{1947}2}+1}=\\=\frac{2^{\frac{1947}2}}{2^{\frac{1947}2}+1}+\frac{1}{2^{\frac{1947}2}+1}=1$$

and so on for the remaining terms.