1. The first term, a, is -49 and the common ratio, r, is 1/7. The sum, S, to n terms is S=a(1-r^n)/(1-r). When n is very large r^n gets very close to zero, so S=a/(1-r)=-49/(6/7)=-49*7/6=-343/6 or -57.1666...7. This is the finite value which the infinite series approaches.
2. You seem to be describing a summation symbol (capital Greek letter sigma, resembling E) between limits for i between 1 and infinity (symbol is like an 8 on its side). This is a shorthand way of writing an infinite geometric series where the general term is 5^(i/2). The first term is 5^(1/2)=sqrt(5) and the common ratio is the same as the first term. The common ratio is bigger than 1 because sqrt(5) is bigger than 2, therefore the series doesn't converge to a finite value, and its sum would be infinite.