For JEE I keep seeing infinite geometric series. Sometimes a formula gives a finite answer and sometimes the series blows up. How do I know which?
An infinite geometric progression with first term a and common ratio r has a finite sum only when the absolute value of r is less than 1, that is when r lies strictly between −1 and 1. In that case the terms shrink toward zero and the sum converges to a divided by (1 − r). If the absolute value of r is greater than or equal to 1, the terms do not shrink, the partial sums grow without bound, and the series diverges, so no finite sum exists. For example 1 + 1/2 + 1/4 + ... has a = 1 and r = 1/2, giving sum 1/(1 − 1/2) = 2. Always check the |r| < 1 condition first before applying the formula, otherwise the answer is meaningless.
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