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Find an example of a continuous function f :[1,)Rsuch that 1fxdxdiverges and localid="1649077247585" k=1fkconverges.

Short Answer

Expert verified

An example issinπx.

Step by step solution

01

Step 1. Given information.

Consider the given question,

The function is [1,)R.

The given integral is1fxdx.

02

Step 2. Evaluate the integral.

Evaluating the integral,

1fxdx=1sinπxdx=limk1ksinπxdx=limk1k-cosπxπk1=limk1k-cosπx+cosππ=limk1k-cosπx-1π=

Thus, the improper integral 1sinπxdxis divergent.

03

Step 3. To prove the series as convergent.

The value of k=1fk=k=1sinπkis given below,

k=1sinπk=sinπ+sin2π+sin3π+...k=1sinπk=0+0+0+...k=1sinπk==0

Thus, the series k=1fk=k=1sinπkis convergent.

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