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Let k=0crkandk=0bvkbe two convergent geometric series. Prove that k=0crk.bvk converges. If neither c nor b is 0, could the series be k=0crkk=0bvk?

Short Answer

Expert verified

k=0crk.bvk=cbk=0rvk

Therefore, the expression k=0crk.bvkis also geometric.

Since rv<1.

Hence it is proven that cbk=0rvkork=0crk.bvkconverges.

And also,k=0crk.bvkk=0crk.k=0bvk.

Step by step solution

01

Step 1. Given Information.

k=0crkandk=0bvkare two convergent geometric series.

02

Step 2. Prove that given expression is geomtric.

When we expand the series k=0crk.bvkwe get,

k=0crk.bvk=cb+cbrv+.....=cb1+(rv)+rv2+.....=cbk=0rvk

As, the above expression is in geometric progression. Therefore the expression k=0crk.bvkis also geometric.

03

Step 3. Prove that it converges.

Now as we know k=0crkconverges which means r<1and also k=0bvkconverges which means v<1.

From this, it can be concluded that rv<1.

Now, look at the expression from previous step we get cbk=0rvk.

And we found that rv<1.

So the expression cbk=0rvkor k=0crk.bvkconverges.

04

Step 4. Proof of last part.

k=0crk.bvk=cb1+rv+rv2+....k=0crk.k=0bvk=c1+r+r2+....b1+v+v2+....

It is clear from above that the right hand side of the above two equations is not equal therefore the left hand sides also cannot be equal as well

Therefore, k=0crk.bvkk=0crk.k=0bvk.

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