List of practice Questions

Consider the two series, \(S_A\) and \(S_B\), where
\[ S_A=\sum_{n=1}^{\infty}\frac{n^2}{2^n} \]
\[ S_B=1+\frac{1}{2}+\frac{1}{8}+\frac{1}{16}+\frac{1}{64}+\frac{1}{128}+\frac{1}{512}+\cdots \]
Which of the following statements is correct for the two given series?

In the Taylor series expansion of function $f(x)=e^{x^{2}-x}$, coefficient of $x^{3}$ is

The Sum $\sum_{r=1}^{20}(r^{2}+1)\times r!$ is equal to

Value of $\sum_{n=0}^{\infty}\frac{2}{(2n+1)(2n+3)}$ is

In a geometric progression, the \(3^{rd}\) term is \(36\) and the \(5^{th}\) term is \(324\). The \(7^{th}\) term of the same progression will be _ _ _. (in integer)

The sequence \(\{a_n = \frac{1}{n^2}; n>0\}\) is

The values of 'm' for which the infinite series,
\(\sum \frac{\sqrt{n+1}+\sqrt{n}}{n^m}\) converges, are:

The limit of the sequence,
\(\{b_n; b_n = \frac{n^n}{(n+1)(n+2)...(n+n)}; n>0\}\), is