Mister Exam

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Sum of series:
a^n
sqrt(n)-2*sqrt(n+1)+sqrt(n+2)
sqrt(1+1)/1
a^2
Identical expressions
one /(n(n+ three)(n+ six))
1 divide by (n(n plus 3)(n plus 6))
one divide by (n(n plus three)(n plus six))
1/nn+3n+6
1 divide by (n(n+3)(n+6))
Similar expressions
1/(n(n+3)(n-6))
1/(n(n-3)(n+6))

Sum of series/ 1/(n(n+3)(n+6))

Sum of series 1/(n(n+3)(n+6))

The solution

You have entered [src]

  oo                   
 ___                   
 \  `                  
  \           1        
   )  -----------------
  /   n*(n + 3)*(n + 6)
 /__,                  
n = 1

\sum_{n=1}^{\infty} \frac{1}{n \left(n + 3\right) \left(n + 6\right)}

Sum(1/((n*(n + 3))*(n + 6)), (n, 1, oo))

The radius of convergence of the power series

Given number:

\frac{1}{n \left(n + 3\right) \left(n + 6\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{1}{n \left(n + 3\right) \left(n + 6\right)}

and

x_{0} = 0

d = 0

c = 1

then

1 = \lim_{n \to \infty}\left(\frac{\left(n + 1\right) \left(n + 4\right) \left(n + 7\right)}{n \left(n + 3\right) \left(n + 6\right)}\right)

Let's take the limit
we find

True

False

The rate of convergence of the power series

The answer [src]

            0                  0   
  -10 + 40*e         -40 + 22*e    
---------------- + ----------------
   /          0\      /          0\
20*\-90 + 90*e /   24*\-45 + 45*e /

\frac{-40 + 22 e^{0}}{24 \left(-45 + 45 e^{0}\right)} + \frac{-10 + 40 e^{0}}{20 \left(-90 + 90 e^{0}\right)}

(-10 + 40*exp_polar(0))/(20*(-90 + 90*exp_polar(0))) + (-40 + 22*exp_polar(0))/(24*(-45 + 45*exp_polar(0)))

Numerical answer [src]

0.0675925925925925925925925925926

0.0675925925925925925925925925926

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 1/(n(n+3)(n+6))

The solution

Examples of finding the sum of a series