A Test for Divergence and Several Tests for Convergence

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A Test for Divergence and Several Tests for Convergence – MCQs

Given that the function ( f ) is positive, decreasing, and continuous for ( x geq 1 ), and each term of the infinite series is defined by ( a_n = f(n) ), identify which statement must necessarily be true if the series ( sum_{n=1}^infty a_n ) converges.

( f(n) ) must tend towards zero as ( n ) approaches infinity.

( f(n) ) can tend towards a positive constant as ( n ) approaches infinity.

The sum of ( f(n) ) over any finite interval can be arbitrarily large.

The function ( f(x) ) can oscillate as ( x ) increases indefinitely.

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