Recognize the pattern in the infinite sum. Observe what changes from the first term to the second. Do this again, observing from the second to the third terms. Do this until you see that the change from one term to the next is clear and static. For example, if you have the infinite sum 1 + x + x^2 + x^3 + x^4 + ..., you will quickly find that each term is the preceding term, multiplied by x.
Write the infinite sum in sigma notation. Sigma notation requires three things: a lower limit, an upper limit and a term. The term is simply the pattern you found in the previous step as a function on "n," which represents the place in the sum. The upper limit will always be infinity for infinite sums. The lower limit is the number that allows the term to match the first term in the sum when substituting "n." For our example, we have the term x^n, which represents x being multiplied n times. The upper limit is infinity. The lower limit is zero, because the first term in the infinite sum is 1 and x^0 = 1.
Rewrite the sigma notation sum as a limit. Change the upper limit into a dummy variable, such as "N." Rewrite the sum as the limit of the sum as N goes to infinity.
Solve the limit. The solution to this limit is the solution to the infinite sum; this alleviates the problem of having to find a sum of an infinite number of terms. For our example, we see that the limit is in the form of a geometric series. Hence, the solution to the limit is 1/(1-x), assuming the absolute value of x is below 1, just as is required by the geometric series.