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Power Series Representation of Functions, Slides of Advanced Calculus

The representation of functions as power series. It covers the concept of power series, the differentiation and integration of power series, and provides examples of finding power series representations for specific functions such as (1 − x)², tan−1 x, and ln (1 + x). The document also includes the radii of convergence for each power series.

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Uploaded on 09/27/2022

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4.6. REPRESENTATION OF FUNCTIONS AS POWER SERIES 99
4.6. Representation of Functions as Power Series
We have already seen that a power series is a particular kind of
function. A slightly different matter is that sometimes a given function
can be written as a power series. We already know the example
1
1x= 1 + x+x2+x3+···+xn+···=
X
n=0
xn(|x|<1)
Replacing xwith other expressions we may write other functions in the
same way, for instance by replacing xwith 2x2we get:
1
1 + 2x2= 12x2+4x48x6+···+(1)n2nx2n+···=
X
n=0
(1)n2nx2n,
which converges for |−2x2|<1, i.e., |x|<1/2.
4.6.1. Differentiation and Integration of Power Series. Since
the sum of a power series is a function we can differentiate it and in-
tegrate it. The result is another function that can also be represented
with another power series. The main related result is that the deriv-
ative or integral of a power series can be computed by term-by-term
differentiation and integration:
4.6.1.1. Term-By-Term Differentiation and Integration. If the power
series P
n=0 cn(xa)nhas radius of convergence R > 0 then the func-
tion
f(x) =
X
n=0
cn(xa)n
is differentiable on the interval (aR, a +R) and
(1) f0(x) =
X
n=0{cn(xa)n}0=
X
n=1
ncn(xa)n1
(2) Zf(x)dx =
X
n=0 Zcn(xa)ndx =C+
X
n=0
cn
(xa)n+1
n+ 1
The radii of convergence of the series in the above equations is R.
Example: Find a power series representation for the function
f(x) = 1
(1 x)2.
pf3

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4.6. Representation of Functions as Power Series

We have already seen that a power series is a particular kind of function. A slightly different matter is that sometimes a given function can be written as a power series. We already know the example

1 1 − x

= 1 + x + x^2 + x^3 + · · · + xn^ + · · · =

∑^ ∞

n=

xn^ (|x| < 1)

Replacing x with other expressions we may write other functions in the same way, for instance by replacing x with − 2 x^2 we get:

1 1 + 2x^2

= 1− 2 x^2 +4x^4 − 8 x^6 +· · ·+(−1)n 2 nx^2 n^ +· · · =

∑^ ∞

n=

(−1)n 2 nx^2 n^ ,

which converges for | − 2 x^2 | < 1, i.e., |x| < 1 /

4.6.1. Differentiation and Integration of Power Series. Since the sum of a power series is a function we can differentiate it and in- tegrate it. The result is another function that can also be represented with another power series. The main related result is that the deriv- ative or integral of a power series can be computed by term-by-term differentiation and integration:

4.6.1.1. Term-By-Term Differentiation and Integration. If the power series

n=0 cn(x^ −^ a)

n (^) has radius of convergence R > 0 then the func-

tion

f (x) =

∑^ ∞

n=

cn(x − a)n

is differentiable on the interval (a − R, a + R) and

(1) f ′(x) =

∑^ ∞

n=

{cn(x − a)n}′^ =

∑^ ∞

n=

ncn(x − a)n−^1

f (x) dx =

∑^ ∞

n=

cn(x − a)n^ dx = C +

∑^ ∞

n=

cn

(x − a)n+ n + 1

The radii of convergence of the series in the above equations is R.

Example: Find a power series representation for the function

f (x) =

(1 − x)^2

Answer : We have 1 (1 − x)^2

d dx

1 − x

and

1 1 − x

∑^ ∞

n=

xn^ ,

hence

1 (1 − x)^2

d dx

∑^ ∞

n=

xn^ =

∑^ ∞

n=

d dx

xn^ =

∑^ ∞

n=

nxn−^1

= 1 + 2x + 3x^2 + 4x^3 + · · · =

∑^ ∞

n=

(n + 1)xn^ (re-indexed)

The radius of convergence is R = 1.

Example: Find a power series representation for tan−^1 x.

Answer : That function is the antiderivative of 1/(1 + x^2 ), hence:

tan−^1 x =

1 + x^2

dx

n=

(−1)nx^2 n^ dx

∑^ ∞

n=

(−1)nx^2 n^ dx

= C +

∑^ ∞

n=

(−1)n^

x^2 n+ 2 n + 1

= C + x −

x^3 3

x^5 5

x^7 7

Since tan−^1 0 = 0 then C = 0, hence

tan−^1 x =

∑^ ∞

n=

(−1)n^

x^2 n+ 2 n + 1

= x −

x^3 3

x^5 5

x^7 7

The radius of convergence is R = 1.

Example: Find a power series representation for ln (1 + x).