The sum of the infinite series 1+23+73^2+123^3+173^4+223^5+ is equal to:

Question

The sum of the infinite series $1+\dfrac23+\dfrac{7}{3^2}+\dfrac{12}{3^3}+\dfrac{17}{3^4}+\dfrac{22}{3^5}+\cdots$ is equal to:

Accepted Answer

Let $S_1 = \dfrac23+\dfrac{7}{3^2}+\dfrac{12}{3^3}+\cdots$ (numerators increasing by $5$). Using the standard subtraction technique ($S_1-\tfrac{S_1}{3}$):

$$\frac{2S_1}{3} = \frac23+\frac{5/3^2}{1-1/3} = \frac23+\frac56 = \frac32 \implies S_1=\frac94.$$

$$S = 1+S_1 = 1+\frac94 = \frac{13}{4}.$$

Solutions · 1