Basicaly by inspection, and making some assumptions.

Look at the first:

1/1 + 3/2 + 5/2^2 + 7/2^3 + ...

The numerators are consecutive odd numbers, which we may write as 2k-1

for k=1, 2, .. The denominators are consecutive powers of 2 which we may

write 2^(k-1) k=1, 2, .. (the -1 in the exponent is required here as I am

taking the first term to correspond to k=1 for both the numerator and

denominator), so the k-th term is (2k-1)/2^{k-1}.

Now look at the second:

1/2 + 2/3 + 3/4 + 4/5 + ...

The numerators are consecutive integers which we may write k, k=1, 2, ..

The denominators are also consecutive integers, but starting from 2, so

may be written (k+1), k=1, 2, .. Hence the general term is k/(k+1).

RonL