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