Mister Exam

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Sum of series 1+1/factorial(n)

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

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

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

The radius of convergence of the power series

Given number:

1 + \frac{1}{n!}

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} = 1 + \frac{1}{n!}

and

x_{0} = 0

d = 0

c = 1

then

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

True

False

The rate of convergence of the power series

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\infty

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