User pie - MathOverflow

pie

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I am a self-learner who aspires to be a mathematician someday. I know I have a lot to learn and I am not very smart, but I will try my best to understand mathematics and achieve my dream.

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0

Is $\sum\limits_{k=1}^nk^i=S_3(n)\times\frac{P_{i-3}(n)}{N_i}$ for odd $i>1,\sum\limits_{k=1}^nk^i=S_2(n)\times\frac{P_{i-2}'(n)}{N_i}$ for even $i$?

23 mins ago

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Conjecture: $\sum\limits_{k=1}^nk^i=S_3(n)\times\frac{P_{i-3}(n)}{N_i}$ for odd $i>1;=S_2(n)\times\frac{P_{i-2}'(n)}{N_i}$ for even $i$.

13

How to evaluate $\int_0^{\frac{\pi}{2}} \frac{\ln(\cos(x))}{1+\sin^2(x)} \, dx$

12

How to solve $\displaystyle\lim_{n\to\infty}\int_0^3\underbrace{\sin(\frac{\pi}{3}\sin(\frac{\pi}{3}...\sin(\frac{\pi}{3} x)...))}_\text{n sines}dx$?

11

How to evaluate $ \sum\limits_{k=0} ^{\infty} \frac{(-1)^k}{4k+3}$?

8

Does real analysis have new theorems, or is it just a collection of proofs of old calculus theorems?