Mister Exam
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How to use it?
- Sum of series:
- 1\3^n
- x^(4n)/n!
- 1/n*(n+1)(n+2)
- nx^n
-
Identical expressions
- one \ three ^n
- 1\3 to the power of n
- one \ three to the power of n
- 1\3n
Sum of series 1\3^n
The solution
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oo ___ \ ` \ -n / 3 /__, n = 1
$$\sum_{n=1}^{\infty} \left(\frac{1}{3}\right)^{n}$$
Sum((1/3)^n, (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\left(\frac{1}{3}\right)^{n}$$
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
$$a_{n} = 1$$
and
$$x_{0} = -3$$
,
$$d = -1$$
,
$$c = 0$$
then
$$\frac{1}{R} = \tilde{\infty} \left(-3 + \lim_{n \to \infty} 1\right)$$
Let's take the limit
we find
$$R = 0$$
$$\left(\frac{1}{3}\right)^{n}$$
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
$$a_{n} = 1$$
and
$$x_{0} = -3$$
,
$$d = -1$$
,
$$c = 0$$
then
$$\frac{1}{R} = \tilde{\infty} \left(-3 + \lim_{n \to \infty} 1\right)$$
Let's take the limit
we find
False
$$R = 0$$
The rate of convergence of the power series
The graph