Sol042706

Evaluate the sum:

$\frac{1}{2-\frac{1}{2}}+\frac{1}{2^2-\frac{1}{2^2}}+ \frac{1}{2^4-\frac{1}{2^4}}+\frac{1}{2^8-\frac{1}{2^8}}+\cdots \frac{1}{2^{2^n}-\frac{1}{2^{2^n}}}+\cdots$

Solution I from landen

It is easily proved by induction on $k$ that:

$\sum_{n=0}^{k}{\frac{1}{x^{2^n}-\frac{1}{x^{2^n}}}}={{1-x^{1-2^{k+1}}}\over{\left(x-1\right)\,\left(1-{{1}\over{x^{2^{k +1}}}}\right)}}$

The question is how to guess to prove that by induction. landen did a few numerical experiments and got, e.g.:

$\sum_{n=0}^{3}{\frac{1}{x^{2^n}-\frac{1}{x^{2^n}}}}=$

${{x^{15}+x^{14}+x^{13}+x^{12}+x^{11}+x^{10}+x^9+x^8+x^7+x^6+x^5+x^4 +x^3+x^2+x}\over{x^{16}-1}}$

Once the pattern was clear it was easy to guess the sum and then prove it with induction. The limit $k\rightarrow\infty$ with $x=2\,$ is $1.\qquad\square$

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