Problems

1. The coefficients in the series \[ S= \tfrac13 x + \tfrac 16 x^2 + \tfrac1{12} x^3 + \cdots + a_rx^r + \cdots \] satisfy a recurrence relation of the form \(a_{r+1} + p a_r =0\). Write down the value of \(p\). By considering \((1+px)S\), find an expression for the sum to infinity of \(S\) (assuming that it exists). Find also an expression for the sum of the first \(n+1\) terms of \(S\).
2. The coefficients in the series \[ T=2 + 8x +18x^2+37 x^3 +\cdots + a_rx^r + \cdots \] satisfy a recurrence relation of the form \(a_{r+2}+pa_{r+1} +qa_r=0\). Find an expression for the sum to infinity of \(T\) (assuming that it exists). By expressing \(T\) in partial fractions, or otherwise, find an expression for the sum of the first \(n+1\) terms of \(T\).

Obtain the sum to infinity of each of the following series.

1. \(1{\displaystyle +\frac{2}{2}+\frac{3}{2^{2}}+\frac{4}{2^{3}}+\cdots+\frac{r}{2^{r-1}}+\cdots;}\)
2. \(1{\displaystyle +\frac{1}{2}\times\frac{1}{2}+\frac{1}{3}\times\frac{1}{2^{2}}+\cdots+\frac{1}{r}\times\frac{1}{2^{r-1}}+\cdots;}\)
3. \({\displaystyle \dfrac{1\times3}{2!}\times\frac{1}{3}+\frac{1\times3\times5}{3!}\frac{1}{3^{2}}+\cdots+\frac{1\times3\times\cdots\times(2k-1)}{k!}\times\frac{1}{3^{k-1}}+\cdots.}\)