Mister Exam
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- Taylor Series Step by Step
- Fourier series step by step
- Differential equations Step by Step
- Product of series
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How to use it?
- Sum of series:
-
1/n^2+2n
- (a)^(i-1)(b)^(k-i)((a)^k+(b)^k)
-
sqrt(n+arctgn^2)-(sqrtn)
- x^i
- Graphing y =:
- 2-x+x^2
-
Identical expressions
- two -x+x^ two
- 2 minus x plus x squared
- two minus x plus x to the power of two
- 2-x+x2
- 2-x+x²
- 2-x+x to the power of 2
-
Similar expressions
- 2-x-x^2
- 2+x+x^2
Sum of series 2-x+x^2
The solution
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oo ___ \ ` \ / 2\ / \2 - x + x / /__, n = 1
$$\sum_{n=1}^{\infty} \left(x^{2} + \left(2 - x\right)\right)$$
Sum(2 - x + x^2, (n, 1, oo))
The radius of convergence of the power series
Given number:
$$x^{2} + \left(2 - x\right)$$
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
$$a_{n} = x^{2} - x + 2$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty} 1$$
Let's take the limit
we find
$$x^{2} + \left(2 - x\right)$$
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
$$a_{n} = x^{2} - x + 2$$
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