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Sum of series 1/(4n-3)(4n+1)

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  oo         
 ___         
 \  `        
  \   4*n + 1
   )  -------
  /   4*n - 3
 /__,        
n = 1

\sum_{n=1}^{\infty} \frac{4 n + 1}{4 n - 3}

Sum((4*n + 1)/(4*n - 3), (n, 1, oo))

The radius of convergence of the power series

Given number:

\frac{4 n + 1}{4 n - 3}

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{4 n + 1}{4 n - 3}

and

x_{0} = 0

d = 0

c = 1

then

1 = \lim_{n \to \infty}\left(\frac{\left(4 n + 1\right)^{2} \left|{\frac{1}{4 n - 3}}\right|}{4 n + 5}\right)

True

False

The rate of convergence of the power series

The answer [src]

oo

\infty

oo

Numerical answer

The series diverges

The graph

Sum of series 1/(4*n-3)(4*n+1)