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Sum of series/ sin(n+1/n)

Sum of series sin(n+1/n)

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  \      /    1\
   )  sin|n + -|
  /      \    n/
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n = 1

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

Sum(sin(n + 1/n), (n, 1, oo))

The radius of convergence of the power series

Given number:

\sin{\left(n + \frac{1}{n} \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} = \sin{\left(n + \frac{1}{n} \right)}

and

x_{0} = 0

d = 0

c = 1

then

1 = \lim_{n \to \infty} \left|{\frac{\sin{\left(n + \frac{1}{n} \right)}}{\sin{\left(n + 1 + \frac{1}{n + 1} \right)}}}\right|

1 = \lim_{n \to \infty} \left|{\frac{\sin{\left(n + \frac{1}{n} \right)}}{\sin{\left(n + 1 + \frac{1}{n + 1} \right)}}}\right|

False

The rate of convergence of the power series

The graph