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Chapter 2: Q10P (page 59)

Find the disk of convergence for each of the following complex power series.

鈭�n=1鈭�(iz)nn2

Short Answer

Expert verified

The required disk of convergence is .|z|<1

Step by step solution

01

Determine Disk of Convergence

For any power series鈭�anzn where z is a complex numbers, then disk of convergence is given by:.蚁=limn鈫�鈭�|z脳nn+1|=|z|

02

Step 2:Find the disk of Convergence

The given power series is:,鈭�n=1鈭�(iz)nn2where, .an=(iz)nn2

Now, let us evaluate the ratio as:

蚁=limn鈫�鈭�|an+1an|=limn鈫�鈭�|(iz)n+1(n+1)2(iz)nn2|=limn鈫�鈭�|iz(nn+1)2|

Now, for the series to be convergent, we have .蚁<1So,

蚁=limn鈫�鈭�|iz(nn+1)2|<1|iz|<limn鈫�鈭�|(nn+1)2||z|<limn鈫�鈭�|(nn+1)2||z|<1

Hence, the required disk of convergence is .|z|<1

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