# What is the Integral Test for Convergence of an Infinite Series?

Integral Test

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The Integral Test is a method used to determine the convergence or divergence of an infinite series. It states that if ( f(x) ) is a continuous, positive, and decreasing function for all ( x \geq 1 ), and if ( a_n = f(n) ) for all positive integers ( n ), then the series ( \sum_{n=1}^{\infty} a_n ) converges if and only if the improper integral ( \int_{1}^{\infty} f(x) , dx ) converges.

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The Integral Test states that if ( f(x) ) is a continuous, positive, and decreasing function for ( x \geq n_0 ) and ( a_n = f(n) ) for all ( n \geq n_0 ), then the series ( \sum_{n=n_0}^{\infty} a_n ) converges if and only if the improper integral ( \int_{n_0}^{\infty} f(x) , dx ) converges.

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