TorioSeduto
New member
- Joined
- Nov 13, 2021
- Messages
- 5
Hello everyone, I have the following number series
[math]\sum_{n=1}^{\infty}{\frac{(7n+32) \cdot 3^{n}}{n(n+2) \cdot 4^{n}}}[/math]
I used the ratio criterion and found that the [math] lim_{n\to \infty} {\Biggl| \frac{a_{n+1}}{a_{n}}\Biggr|} \,=\, \frac{3}{4}[/math]
Knowing that the series converges, at this point, to calculate the sum of the series, I thought of using the partial fraction decomposition and I rewrote the series as: [math]\sum_{n=1}^{\infty}{\Biggl( \frac{16}{n} - \frac{9}{n+2}\Biggr) \cdot \Biggr(\frac{3}{4}\Biggr)^{n}}[/math]
At this point how can I calculate the sum of the series?
[math]\sum_{n=1}^{\infty}{\frac{(7n+32) \cdot 3^{n}}{n(n+2) \cdot 4^{n}}}[/math]
I used the ratio criterion and found that the [math] lim_{n\to \infty} {\Biggl| \frac{a_{n+1}}{a_{n}}\Biggr|} \,=\, \frac{3}{4}[/math]
Knowing that the series converges, at this point, to calculate the sum of the series, I thought of using the partial fraction decomposition and I rewrote the series as: [math]\sum_{n=1}^{\infty}{\Biggl( \frac{16}{n} - \frac{9}{n+2}\Biggr) \cdot \Biggr(\frac{3}{4}\Biggr)^{n}}[/math]
At this point how can I calculate the sum of the series?