Solved Problems about Power Series

1
Solved Problems about Power Series

• Summary of Power Series
• Overview of Problems
• Radius of Convergence Problems
• Convergence of the Differentiated and the
  Integrated Power Series
• Finding Power Series
• Fibonacci Power Series
• Indefinite integrals of Power Series
• Definite Integrals of Power Series

2
Summary of Power Series
Power Series
Basic Power Series
By the Ratio Test
Radius of Convergence
By the Root Test
The Power Series S(x) converges if x lt R and
diverges if x gt R.
Differentiation
Integration
3
Overview of Problems
1
2
3
4
5
4
Overview of Problems
6
7
8
9
5
Radius of Convergence
1
Solution
6
Radius of Convergence
2
Solution
7
Converges of the Differentiated Series
3
Solution
Next use the fact that the limit of a product is
the product of the limits.
8
Convergence of the Integrated Series
4
Solution
Next use the fact that the limit of a product is
the product of the limits.
9
Finding Power Series for Functions
5
Solution
Multiply by x2 to get
The Radius of Convergence is
Observe that the function is undefined for x ½.
Hence it is to be expected that a power series
for the function will not converge for x ½.
10
Fibonacci Power Series
6
Solution
11
Finding Power Series
7
Solution
Substitute t -x to the above power series in t
to get
Multiply by x2 to get
12
Power Series for Indefinite Integrals
8
Solution
Here C is the constant of integration.
Substitute x -t5 to the above power series for
1/(1-x) to get
Integrate the power series term by term to get
13
Power Series for Definite Integrals
9
Solution
Integrate this series.
This approximation is already accurate enough
since the series is alternating and the first
term left out is 1/22528 lt 0.001.