# How can an infinite series have a finite sum?

Yes, the same question used to bother me as well, but the following example might make you feel more comfortable with it.

Are you now convinced that an infinite series can have a finite sum?

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An infinite series can have a finite sum if the terms of the series approach zero sufficiently quickly as the series progresses. This concept is based on the idea of convergence in mathematics.

Mathematically, an infinite series (\sum_{n=1}^{\infty} a_n) converges to a finite sum (S) if the sequence of partial sums (S_N = \sum_{n=1}^{N} a_n) approaches (S) as (N) approaches infinity.

Several tests are used to determine the convergence of infinite series, such as the ratio test, root test, and comparison test. If these tests confirm that the series converges, then its sum is finite.

For example, the geometric series ( \sum_{n=0}^{\infty} r^n ) converges to ( \frac{1}{1-r} ) if ( |r| < 1 ), even though it has infinitely many terms. This is because as more terms are added, each term becomes smaller, and the sum approaches a finite value.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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- How does an infinite geometric series apply to being pushed on a swing?

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