0

Query 1

Using the formula

$$ \sum_{j=1}^{n} \sin(jt) = \frac{\cos\frac{t}{2}-\cos\left(\left(n+\frac{1}{2}\right)t\right)}{2\sin\frac{t}{2}} $$

evaluate $\int_{0}^{\frac{\pi}{2}} \sin(x) \space dx$ as the limit of a Riemann Sum.

$\underline{Attempt}$

\begin{align} \int_{0}^{\frac{\pi}{2}} \sin(x) \space dx &= \frac{\pi}{2n} \lim_{n\to \infty} \sum_{j=1}^{n} \sin\left(j\frac{\pi}{2n}\right) \leftarrow(\textit {Let $t$ = $\frac{\pi}{2n}$)} \\ &= \frac{\pi}{2n} \lim_{n\to \infty} \frac{\cos\frac{\pi}{4n}-\cos\left(\left(n+\frac{1}{2}\right)\left(\frac{\pi}{2n}\right)\right)}{2\sin\frac{\pi}{4n}} \\ &= \lim_{n\to \infty} \left(\frac{\pi}{2n}\right)\frac{\cos\frac{\pi}{4n}-\cos\left(\frac{\pi}{2}+\frac{\pi}{4n}\right)}{2\sin\frac{\pi}{4n}} \\ &= \lim_{n\to \infty} \left(\frac{\pi}{2n}\right)\frac{\cos\frac{\pi}{4n}-\sin\frac{\pi}{4n}}{2\sin\frac{\pi}{4n}} \end{align}

Query 2

Using the result

$ \forall k \in \mathbb Z^+ ,$ $$ \\ \sum_{j=1}^{n} j^k = \frac{n^{k+1}}{k+1}+\frac{n^k}{2}+P_{k-1}(n), $$ where $P_{k-1}$ is a polynomial of degree at most $k-1$, deduce the result: $$ \int_{0}^{a} x^k \space dx = \frac{a^{k+1}}{k+1} $$

$\underline{Comment}$ : I do not understand why only the first portion of the summation result is used.

Thanks!

NetUser5y62
  • 966
  • Use \begin{align} and \end{align}. and use & on that part of the line where you want to align... – King Tut Mar 13 '18 at 13:32
  • @KingTut: Thanks! Previously I used &=& to align at the equal sign, instead of &= like you recommended. This used to work in Latex, but not here on Mathjax. – NetUser5y62 Mar 13 '18 at 13:37
  • Netuser I think first you need to find the sum of sines whose phase are in arithmetic progression. Consider imaginary part of $e^ {ix}$ – King Tut Mar 13 '18 at 13:43

1 Answers1

2

Hint for Query 1: $$ \left(\frac\pi{2n}\right)\frac{\cos\left(\frac\pi{4n}\right)-\sin\left(\frac\pi{4n}\right)}{2\sin\left(\frac\pi{4n}\right)} =\frac{\overbrace{\cos\left(\frac\pi{4n}\right)}^{\to?}-\overbrace{\sin\left(\frac\pi{4n}\right)\,}^{\to?}}{\underbrace{\dfrac{\sin\left(\frac\pi{4n}\right)}{\frac\pi{4n}}}_{\to?}} $$ For the denominator, see this answer.

Hint for Query 2: $$ \int_0^ax^k\,\mathrm{d}x\ \ \overset{\text{Riemann Sum}}\longrightarrow\ \ \sum_{j=1}^n\left(\frac{aj}n\right)^k\frac an $$

robjohn
  • 345,667