Infinite series: the range of x, r and the first term Problem set A

Find the sum to infinity of each series.

1. (a) \(\frac23-\frac29+\frac{2}{27}-\dots\)
2. (b) \(24-18+\frac{27}{2}-\dots\)

Find the range of values of x for which each series converges.

1. (a) \((x-6)+(x-6)^2+(x-6)^3+\dots\)
2. (b) \(1+\frac x4+\frac{x^2}{16}+\dots\)
3. (c) \(1-\frac{5x}{3}+\frac{25x^2}{9}-\dots\)

Find the range of values of x for which each series converges.

1. (a) \(1+5x^2+25x^4+\dots\)
2. (b) \(1+\frac2x+\frac4{x^2}+\dots\)

A geometric series has first term 15 and sum to infinity 40. Find the common ratio.

A geometric series has sum to infinity 35 and common ratio \(-\frac{2}{5}\). Find the first term, and write down the first three terms.

A series is \(\sum_{k=1}^{\infty}12\left(\frac13\right)^k\).

1. (a) Write down the first three terms.
2. (b) Find the sum to infinity.

A geometric series has second term 8 and sum to infinity 50.

1. (a) Find the two possible values of r.
2. (b) Find the two possible values of \(u_1\).

A series is \(4+4\left(\frac{x-1}{3}\right)+4\left(\frac{x-1}{3}\right)^2+\dots\)

1. (a) Find the range of values of x for which the series converges.
2. (b) Find an expression for \(S_\infty\) in terms of x.
3. (c) Find \(S_\infty\) when \(x=2\).

The sum to infinity of \(\sum_{k=1}^{\infty}5\left(\frac x4\right)^{k-1}\) is 8. Find the value of x.

A trust will pay out a total of 50 000 euros over an unlimited number of years. It pays 12 000 euros in the first year, and each later year pays a constant fraction r of the year before. Find r.

The weekly water supplied by a well forms a geometric sequence. Over all the weeks to come it will supply 54 thousand litres, and in the first three weeks it supplies 38 thousand litres. Find the volume, in thousand litres, supplied in the first week.

A geometric series has second term \(-5\) and sum to infinity 16.

1. (a) Find the possible values of r. Reject any that cannot be used.
2. (b) Find \(u_1\).