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1/4n^2-1

Sum of series 1/4n^2-1



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The solution

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  oo          
____          
\   `         
 \    / 2    \
  \   |n     |
  /   |-- - 1|
 /    \4     /
/___,         
n = 1         
$$\sum_{n=1}^{\infty} \left(\frac{n^{2}}{4} - 1\right)$$
Sum(n^2/4 - 1, (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\frac{n^{2}}{4} - 1$$
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
$$a_{n} = \frac{n^{2}}{4} - 1$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty} \left|{\frac{\frac{n^{2}}{4} - 1}{\frac{\left(n + 1\right)^{2}}{4} - 1}}\right|$$
Let's take the limit
we find
True

False
The rate of convergence of the power series
Numerical answer
The series diverges
The graph
Sum of series 1/4n^2-1
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