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- Power Series Definition. A power series, centered at a, is of the form
∑^ ∞
n=
cn(x − a)n^ = c 0 + c 1 (x − a) + c 2 (x − a)^2 + · · · ,
where a is a fixed number, x is a variable. cn’s are constants, called the coefficients of the series.
Example. When a =0, then a power series is of the form ∑^ ∞
n=
cnxn^ = c 0 + c 1 x + c 2 x^2 + · · ·.
This resembles a polynomial, except it has infinitely many terms.
For a fixed x, we have a series and we can check for convergence or divergence. A power series may converge for some values of x and diverge for other values of x.
Question 1. Consider the series (^) ∞ ∑
n=
(x − 1)n (n + 1)
a) Check that the series converges for x = 0.
b) Check that the series diverges for x = 2.
1
Definition. A power series
f (x) =
∑^ ∞
n=
cn(x − a)n
represents a function of x, who domain is the set of all x for which the series converges.
Note that the series always converges for x = a, since f (a) = c 0.
Question 2. For what values of x does the series ∑^ ∞
n=
(x − 3)n n
convergent.
Hint: Use the Ratio Test, and you need to consider the endpoints seperately.
Question 4. Find the radius and interval of convergence of the series
∑^ ∞
n=
(n + 1)!(x − 4)n
- Series Representation of the function We know that a power series represents a function of x, whose domain is the interval of convergence of the series. Now we want to work the other way round, and express a function as a power series.
Example. Consider the power series ∑^ ∞
n=
xn^ = 1 + x + x^2 + · · ·.
It is a geometric series with first term a = 1 and r = x. Then we know that the series
converges to 1 1 − x
, when − 1 < x < 1.
We say that this series represents the function f (x) = 1 1 − x
in the interval − 1 < x < 1, and
we write
1 1 − x
∑^ ∞
n=
xn^ − 1 < x < 1
Question 5. Find a power series representation for f (x) = 1 x + 3
Any function that can be represented by a power series in x − c with a positive or infinite radius of convergence (R > 0 or R = ∞) is said to be analytic at c. For example, the function
f (x) = 1 1 − x
is analytic at x = 0. A function that is analytic at x = c has derivatives of all
orders at x = c, i.e. f (n)(c) exists ffor all positive integers n.
Question 6. Find the power series representation of f (x) = x
3 x + 3
. Hint: Use your answer
from Question 5.
Question 8. Find the power series representation for ln(1 + x) and its radius of convergence.
Question 9. a) Find a power series representation for f (x) = arctan x.
b) Use your answer in part a) to show that
π = 2
∑^ ∞
n=
(−1)n (2n + 1) 3n
