Question

# Find the sum of the series: sum_{n=0}^infty((frac{5}{2^n})-(frac{1}{3^n}))

Series
Find the sum of the series:
$$\sum_{n=0}^\infty((\frac{5}{2^n})-(\frac{1}{3^n}))$$

2021-01-20
The given series is
$$\sum_{n=0}^\infty((\frac{5}{2^n})-(\frac{1}{3^n}))$$
Rewrite the given series as shown below:
$$\sum_{n=0}^\infty((\frac{5}{2^n})-(\frac{1}{3^n}))=\sum_{n=0}^\infty(\frac{5}{2^n})-\sum_{n=0}^\infty(\frac{1}{3^n})$$
From above it can be observed that given series is the sum of two geometric series. Hence, the sum of series is obtained as,
$$\sum_{n=0}^\infty((\frac{5}{2^n})-(\frac{1}{3^n}))=\sum_{n=0}^\infty(\frac{5}{2^n})-\sum_{n=0}^\infty(\frac{1}{3^n})$$
$$=5\sum_{n=0}^\infty(\frac{1}{2^n})-\sum_{n=0}^\infty(\frac{1}{3^n})$$
$$=5(\frac{\frac12}{1-\frac12})-(\frac{\frac13}{1-\frac13})$$
$$=5(\frac{\frac12}{\frac12})-(\frac{\frac13}{\frac23})$$
$$=5(1)-(\frac12)$$
$$=\frac92$$