New posts in Sequences And Series

Variant of Parseval equality in Hilbert space

Integral $\int_0^1(1-x^3+x^5-x^8+x^{10}-x^{13}+\dots)dx$

prove that $ \sin(x+y)=\sin{x} \cdot \cos{y} +\sin{y} \cdot \cos{x} $ using power series (without trigonometric identities)

Prove that the sequence $(a_n/n)$ converges

Periodic sequence problem

Show that $\lim_{n \to \infty}\frac {a_n} {n}=1 \implies \lim_{n \to \infty }\sup_{0\le k \le n} \frac {|a_k - k|} {n }=0$

Proof that $\sum_{n=0}^{\infty}(-1)^{n} = \frac{1}{2}$. Is there any error?

Let $a_n>0$; $\sum a_n$ diverges; find $b_n$ s.t. $b_n>0$; $b_n/a_n\to0$; $\sum b_n$ diverges

Does the fraction of distinct substrings in prefixes of the Thue–Morse sequence of length $2^n$ tend to $73/96$?

Finding the Sum of series $S = \sum_{n=1}^{243} \frac{1}{n^{4/5}} $.