4 General power series
A general power series has the form
Exactly the same considerations apply to this general power series as apply to the ‘special’ series except that the variable is replaced by . The radius of convergence of the general series is obtained in the same way:
and the interval of convergence is now shifted to have centre at (see Figure 4 below). The series is absolutely convergent if , diverges if and may or may not converge if
Figure 4
Task!
Find the radius of convergence of the general power series
First find an expression for the general term:
Now obtain the radius of convergence:
Hence
, so the series is absolutely convergent if
.
Finally, decide on the convergence at (i.e. at and i.e. and ):
At the series is which diverges and at the series is which also diverges. Thus the given series only converges if i.e. .
Exercises
-
From the result
- Find an expression for
- Use this expression to obtain an approximation to to 4 d.p.
- Find the radius of convergence of the general power series
- Find the range of values of for which the power series converges.
- By differentiating the series for find the power series for and state its radius of convergence.
-
- Find the radius of convergence of the series
- Investigate what happens at the points and
- (4 d.p.)
- . Series converges if . If series diverges. If series diverges.
- Series converges if .
- valid for .
-
- .
- At series diverges. At series converges.