find the sum of 1/3 - $3/2^{2}$ + $5/2^{3}$ .... $\infty$
You asked:
Find the sum of the sequence: \(\frac{1}{3}\), \(-\left( \frac{3}{{2}^{2}} \right)\), \(\frac{5}{{2}^{3}}\), \(\ldots\), \(\infty\)
MathBot Answer:
The infinite series \(\displaystyle \sum_{n=1}^{\infty} \left(- \frac{2197 \left(- \frac{33}{26}\right)^{n}}{5841} - \frac{17}{118}\right)\) diverges.
This is a recurrent 2 sequence.
The nth term in this sequence is given by the formula:
Explicit Formula: $$a_n=- \frac{2197 \left(- \frac{33}{26}\right)^{n}}{5841} - \frac{17}{118}$$
Recursive Formula: $$a_n=- \frac{33 a_{n-1}}{26} - \frac{17}{52}, \text{where } a_1=\frac{1}{3}$$