# Does $1+\frac{1}{1^k+2^k}+\frac{1}{1^k+2^k+3^k}+\frac{1}{1^k+2^k+3^k+4^k}+\frac{1}{1^k+2^k+3^k+4^k+5^k}\cdots$ telescope?

after the post $1+\frac{1}{1+2}+\frac{1}{1+2+3}+\frac{1}{1+2+3+4}+\frac{1}{1+2+3+4+5}+\cdots$, I had to ask

Can we telescope the series $$1+\frac{1}{1^k+2^k}+\frac{1}{1^k+2^k+3^k}+\frac{1}{1^k+2^k+3^k+4^k}+\frac{1}{1^k+2^k+3^k+4^k+5^k}+\frac{1}{1^k+2^k+3^k+4^k+5^k+6^k}+\cdots$$ ?

-