Mister Exam

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How to use it?
Sum of series:
1/((3*n+1)*(3*n+4))
(2n+1)/(n(n+2)(n+4))
1/(2*n-1)*(2*n+1)
ln(n+1)/n
Identical expressions
(two n+ one)/(n(n+2)(n+ four))
(2n plus 1) divide by (n(n plus 2)(n plus 4))
(two n plus one) divide by (n(n plus 2)(n plus four))
2n+1/nn+2n+4
(2n+1) divide by (n(n+2)(n+4))
Similar expressions
(2n+1)/(n(n-2)(n+4))
(2n-1)/(n(n+2)(n+4))
(2n+1)/(n(n+2)(n-4))

Sum of series/ (2n+1)/(n(n+2)(n+4))

Sum of series (2n+1)/(n(n+2)(n+4))

The solution

You have entered [src]

  oo                   
 ___                   
 \  `                  
  \        2*n + 1     
   )  -----------------
  /   n*(n + 2)*(n + 4)
 /__,                  
n = 1

\sum_{n=1}^{\infty} \frac{2 n + 1}{n \left(n + 2\right) \left(n + 4\right)}

Sum((2*n + 1)/(((n*(n + 2))*(n + 4))), (n, 1, oo))

The radius of convergence of the power series

Given number:

\frac{2 n + 1}{n \left(n + 2\right) \left(n + 4\right)}

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{2 n + 1}{n \left(n + 2\right) \left(n + 4\right)}

and

x_{0} = 0

d = 0

c = 1

then

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

Let's take the limit
we find

True

False

The rate of convergence of the power series

The answer [src]

               0               0   
7      -1 + 4*e        -7 + 3*e    
-- + ------------- + --------------
12     /        0\      /        0\
     9*\-8 + 8*e /   12*\-8 + 8*e /

\frac{-7 + 3 e^{0}}{12 \left(-8 + 8 e^{0}\right)} + \frac{7}{12} + \frac{-1 + 4 e^{0}}{9 \left(-8 + 8 e^{0}\right)}

7/12 + (-1 + 4*exp_polar(0))/(9*(-8 + 8*exp_polar(0))) + (-7 + 3*exp_polar(0))/(12*(-8 + 8*exp_polar(0)))

Numerical answer [src]

0.697916666666666666666666666667

0.697916666666666666666666666667

The graph

Examples of finding the sum of a series

The Sum of the Power Series
```
x^n/n
```
```
(x-1)^n
```
Factorial
```
1/2^(n!)
```
```
n^2/n!
```
```
x^n/n!
```
```
k!/(n!*(n+k)!)
```
Flint Hills Series
```
csc(n)^2/n^3
```
Basel problem
```
1/n^2
```
```
1/n^4
```
```
1/n^6
```
Harmonic series
```
1/n
```
Grandi's series
```
(-1)^n
```
Alternating series
```
(-1)^(n + 1)/n
```
```
(n + 2)*(-1)^(n - 1)
```
```
(3*n - 1)/(-5)^n
```
```
(-1)^(n - 1)*n/(6*n - 5)
```
Newton–Mercator series
```
(-1)^(n + 1)/n*x^n
```
Exam the series for convergence
```
(3*n - 1)/(-5)^n
```

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How to use it?

Identical expressions

Similar expressions

Sum of series (2n+1)/(n(n+2)(n+4))

The solution

Examples of finding the sum of a series