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Real analysis

Show that if sum(sum sign) to infinity(top) of k=1(bottom) of a_k=A and sum(sum sign) to infinity(top) of k=1(bottom) of b_k=B, then
1-sum(sum sign) to infinity(top) of k=1(bottom) of ca_k=cA for all c belong to R
2-sum(sum sign) to infinity(top) of k=1(bottom) of (a_k+b_k)=A+B

Solution Preview

1)
We know that the series converges to A. That means
lim s_k= lim (a_1 + a_2 + ... + a_k) = A
k--->infinity

Now we know from the standard laws of lim that if:

lim (f(x))= L
x--->h
then ...

Solution Summary

There are several proofs regarding summations in this solution.

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