Mister Exam

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Sum of series:
t^k*(-1)^k*a^(2*k)*cos(a*x)*e^(a*z)/factorial(k)
5/(25*n^2-5*n-6)
tan(sqrt(n)/(n^2+1))
arctan(n+3)-arctan(n+1)
Identical expressions
arctan(n+ three)-arctan(n+ one)
arc tangent of (n plus 3) minus arc tangent of (n plus 1)
arc tangent of (n plus three) minus arc tangent of (n plus one)
arctann+3-arctann+1
Similar expressions
arctan(n-3)-arctan(n+1)
arctan(n+3)-arctan(n-1)
arctan(n+3)+arctan(n+1)

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

Sum of series arctan(n+3)-arctan(n+1)

The solution

You have entered [src]

  oo                             
 __                              
 \ `                             
  )   (atan(n + 3) - atan(n + 1))
 /_,                             
n = 1

\sum_{n=1}^{\infty} \left(- \operatorname{atan}{\left(n + 1 \right)} + \operatorname{atan}{\left(n + 3 \right)}\right)

Sum(atan(n + 3) - atan(n + 1), (n, 1, oo))

The radius of convergence of the power series

Given number:

- \operatorname{atan}{\left(n + 1 \right)} + \operatorname{atan}{\left(n + 3 \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} = - \operatorname{atan}{\left(n + 1 \right)} + \operatorname{atan}{\left(n + 3 \right)}

and

x_{0} = 0

d = 0

c = 1

then

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

Let's take the limit
we find

True

False

The rate of convergence of the power series

The answer [src]

pi - atan(2) - atan(3)

- \operatorname{atan}{\left(3 \right)} - \operatorname{atan}{\left(2 \right)} + \pi

pi - atan(2) - atan(3)

Numerical answer [src]

0.785398163397448309615660845820

0.785398163397448309615660845820

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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Identical expressions

Similar expressions

Sum of series arctan(n+3)-arctan(n+1)

The solution

Examples of finding the sum of a series