Mister Exam

Other calculators

8/(16n^2+8n-15)
Sum of series/ 8/(16n^2+8n-15)

Sum of series 8/(16n^2+8n-15)


∑
=

The solution

You have entered [src] 
  oo                  
____                  
\   `                 
 \           8        
  \   ----------------
  /       2           
 /    16*n  + 8*n - 15
/___,                 
n = 1
$$\sum_{n=1}^{\infty} \frac{8}{\left(16 n^{2} + 8 n\right) - 15}$$ 
Sum(8/(16*n^2 + 8*n - 15), (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\frac{8}{\left(16 n^{2} + 8 n\right) - 15}$$
It is a series of species
$$a_{n} \left(c x - x_{0}\right)^{d n}$$
- power series.
The radius of convergence of a power series can be calculated by the formula:
$$R^{d} = \frac{x_{0} + \lim_{n \to \infty} \left|{\frac{a_{n}}{a_{n + 1}}}\right|}{c}$$
In this case
$$a_{n} = \frac{8}{16 n^{2} + 8 n - 15}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\left(8 n + 16 \left(n + 1\right)^{2} - 7\right) \left|{\frac{1}{16 n^{2} + 8 n - 15}}\right|\right)$$
Let's take the limit
we find
True

False
The rate of convergence of the power series
The answer [src] 
8*Gamma(13/4)
-------------
15*Gamma(9/4)
$$\frac{8 \Gamma\left(\frac{13}{4}\right)}{15 \Gamma\left(\frac{9}{4}\right)}$$ 
8*gamma(13/4)/(15*gamma(9/4))
Numerical answer [src] 
1.20000000000000000000000000000
1.20000000000000000000000000000
The graph
Sum of series 8/(16n^2+8n-15)

    Examples of finding the sum of a series

    © Mister Exam – Calculator