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Tag Archives: Real Analysis
Let denote the fractional part. Prove that
for the different values of the integer number .
Let denote the integral,
since if whereas if . Therefore,
except of a countable set whose measure is .
Let denote the Riemann zeta function. Evaluate the integral
Based on symmetries,
Let . It follows that
Using the recursion we get that
Let denote the trigamma function and let be integrable on . It holds that
We have successively:
where the interchange between the infinite sum and the integration is allowed by the uniform bound
The result follows.
Let denote the digamma series. Evaluate the series
The series telescopes;
Let be a continuous real-valued function on satisfying
Define a function on by
- Prove that is continuous and periodic with period .
- Prove that if is continuous and periodic with period then
- We note that
- First of all is bounded on and uniformly. Hence,