Chapter 9

Sum the series \(n \cdot 1+(n-1) \cdot 2+(n-2) \cdot 3+\cdots \cdots+1 \cdot n\)

Find the sum of all possible products of the first \(n\) natural numbers taken two by two.

Find the coefficient of \(x^{99}\) in the polynomial \((x-1)(x-2)(x-3) \cdots \cdots(x-100)\)

If \(s\) and \(t\) are respectively the sum and the sum of the squares of \(n\) successive positive integers beginning with \(a\) then show that \(n t-s^{2}\) is independent of \(a\).

If \(S_{1}, S_{2}, S_{3}, \ldots, S_{n}\) are the sums of infinite geometric series whose first terms are \(1,2,3, \ldots \ldots, n\) and whose common ratios are \(\frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \ldots, \frac{1}{n+1}\) respectively, then find the value of \(S_{1}^{2}+S_{2}^{2}+S_{3}^{2}+\cdots \cdots+S_{2 n-1}{\underline{\phantom{xx}}}^{2}\).

Consider \(n\) A.P.s whose first terms are \(1,2,3, \ldots \ldots \ldots, n\) and their common differences are \(1,2,3, \ldots \ldots \ldots .\) \(n\) respectively. If \(S_{i, i}\) denotes sum of \(i\) terms of \(i^{\text {th }}\) A.P., then find \(S_{1,1}+S_{2,2}+S_{3,3}+\cdots \cdots+S_{n, n} \cdot\\{\)

On the ground are placed \(n\) stones, the distance between the first and second is one yard, between the 2 nd and 3rd is 3 yards, between the 3rd and 4 th, 5 yards, and so on. How far will a person have to travel who shall bring them one by one to a basket placed at the first stone?

Find \(\lim _{n \rightarrow \infty}\left\\{\frac{1}{1-n^{2}}+\frac{2}{1-n^{2}}+\frac{3}{1-n^{2}}+\cdots \cdots+\frac{n}{1-n^{2}}\right\\} .\)

Find \(\lim _{n \rightarrow \infty} \frac{1^{2}+2^{2}+3^{2}+\cdots \cdots+n^{2}}{n^{3}}\).

Sum the series \(1+3+6+10+15+\ldots\) to \(n\) terms.

Sum the series \(2+4+7+11+16+\ldots .\) to \(n\) terms.

Sum up to \(n\) terms the series \(0.7+0.77+0.777+\ldots\)

Sum up to \(n\) terms the series \(6+66+666+\ldots\)

Sum the series \(1+2.2+3.2^{2}+4.2^{3}+\ldots+100.2^{99}\).

Find the sum of \(n\) terms of the series the \(r\) th term of which is \((2 r+1) 3^{r}\).

Sum the series \(1+\frac{3}{2}+\frac{5}{4}+\frac{7}{8}+\ldots n\) terms.

Sum the series \(1+\frac{4}{5}+\frac{7}{5^{2}}+\frac{10}{5^{3}}+\ldots\) to \(n\) terms and to \(\infty\).

Sum the series \(1+\frac{1}{3}+\frac{3}{3^{2}}+\frac{5}{3^{3}}+\ldots\) to \(\infty\)

Sum to \(n\) terms the series and sum of infinite series \(\frac{1}{1 \cdot 2}+\frac{1}{2 \cdot 3}+\frac{1}{3 \cdot 4}+.\)

Sum to \(n\) terms the series and sum of infinite series \(\frac{1}{3 \cdot 5}+\frac{1}{5 \cdot 7}+\frac{1}{7 \cdot 9}+\ldots\)

Sum to \(n\) terms the series and sum of infinite series \(1+\frac{1}{1+2}+\frac{1}{1+2+3}+\ldots\)

Sum to \(n\) terms the series and sum of infinite series \(\frac{1}{1 \cdot 3 \cdot 5}+\frac{1}{3 \cdot 5 \cdot 7}+\frac{1}{5 \cdot 7 \cdot 9}+\ldots\)

Sum to \(n\) terms the series and sum of infinite series \(\frac{3}{1^{2} \cdot 2^{2}}+\frac{5}{2^{2} \cdot 3^{2}}+\frac{7}{3^{2} \cdot 4^{2}}+\ldots\)

Sum to \(n\) terms the series and sum of infinite series \(\frac{1}{1 \cdot 3}+\frac{2}{1 \cdot 3 \cdot 5}+\frac{3}{1 \cdot 3 \cdot 5 \cdot 7}+\ldots\)

Sum to \(n\) terms the series and sum of infinite series \(\frac{1}{\sqrt{2}+\sqrt{1}}+\frac{1}{\sqrt{3}+\sqrt{2}}+\frac{1}{\sqrt{4}+\sqrt{3}}+\).

Sum to \(n\) terms the series and sum of infinite series \(\ln \left(1-\frac{1}{2^{2}}\right)+\ln \left(1-\frac{1}{3^{2}}\right)+\ln \left(1-\frac{1}{4^{2}}\right)+\ldots\)

Find \(\lim _{n \rightarrow \infty} \sum_{r=1}^{n} \frac{1}{(r+2) \cdot r !} .\)

Prove that \(1-\frac{1}{n+1}<\frac{1}{1^{2}}+\frac{1}{2^{2}}+\frac{1}{3^{2}}+\cdots \cdots+\frac{1}{n^{2}}<2-\frac{1}{n}\).

Find \(\sum_{n=1}^{\infty} \frac{1}{(n+1) !}\)

Find \(\sum_{n=1}^{\infty} \frac{1}{(n+2)} .\)

Find \(\sum_{n=1}^{\infty} \frac{1}{(2 n-1) !}\)

Find \(\sum_{n=1}^{\infty} \frac{1}{(2 n+1)} .\)

Sum the infinite series \(1+\frac{3}{1 !}+\frac{5}{2 !}+\frac{7}{3 !}+\cdots \cdot\)

Sum the infinite series \(1+\frac{2^{2}}{2 !}+\frac{3^{2}}{3 !}+\frac{4^{2}}{4 !}+\).

Sum the infinite series \(1+\frac{2^{3}}{2 !}+\frac{3^{3}}{3 !}+\frac{4^{3}}{4 !}+\cdots\)

Sum the infinite series \(\frac{2}{3 !}+\frac{4}{5 !}+\frac{6}{7 !}+\frac{8}{9 !}+\cdots\)

Sum the infinite series \(\frac{2}{1 !}+\frac{4}{3 !}+\frac{6}{5 !}+\frac{8}{7 !}+\).

Sum the infinite series \(1+\frac{1+a}{2 !}+\frac{1+a+a^{2}}{3 !}+\frac{1+a+a^{2}+a^{3}}{4 !}+\cdots\)

Sum the infinite series \(1+\frac{1+2}{2 !}+\frac{1+2+3}{3 !}+\frac{1+2+3+4}{4 !}+\cdots \cdots\)

Sum the infinite series \(\frac{1}{2 !}+\frac{1+2}{3 !}+\frac{1+2+3}{4 !}+\cdots \cdots\)

Sum the infinite series \(\frac{1^{2} \cdot 2}{1 !}+\frac{2^{2} \cdot 3}{2 !}+\frac{3^{2} \cdot 4}{3 !}+\cdots\)

Sum the infinite series \(1+\frac{3}{2 !}+\frac{6}{3 !}+\frac{10}{4 !}+\cdots\)

Sum the infinite series \(\frac{12}{2 !}+\frac{28}{3 !}+\frac{50}{4 !}+\frac{78}{5 !}+\cdots\)

Sum the infinite series \(\frac{1}{3}+\frac{1}{3 \cdot 3^{3}}+\frac{1}{5 \cdot 3^{5}}+\frac{1}{7 \cdot 3^{7}}+\cdots\)

Sum the infinite series \(1+\frac{1}{3 \cdot 2^{2}}+\frac{1}{5 \cdot 2^{4}}+\frac{1}{7 \cdot 2^{6}}+\cdots\)

Sum the infinite series \(\frac{1}{1 \cdot 2}+\frac{1}{3 \cdot 4}+\frac{1}{5 \cdot 6}+\cdots \cdot\)

Sum the infinite series \(\frac{1}{2 \cdot 3}+\frac{1}{4 \cdot 5}+\frac{1}{6 \cdot 7}+\cdots \cdots\)

Sum the infinite series \(\frac{1}{1 \cdot 3}+\frac{1}{2 \cdot 5}+\frac{1}{3 \cdot 7}+\frac{1}{4 \cdot 9}+\cdots \cdot\)

Sum the infinite series \(\frac{5}{1 \cdot 2 \cdot 3}+\frac{7}{3 \cdot 4 \cdot 5}+\frac{9}{5 \cdot 6 \cdot 7}+\cdots\)

Sum the infinite series \(\frac{1}{1 \cdot 2 \cdot 3}+\frac{1}{3 \cdot 4 \cdot 5}+\frac{1}{5 \cdot 6 \cdot 7}+\cdots\)