Integral test for convergence

In mathematics, the integral test for convergence is a method used to test infinite series of non-negative terms for convergence. It was developed by Colin Maclaurin and Augustin-Louis Cauchy and is sometimes known as the Maclaurin–Cauchy test. ∫ N ∞ f ( x ) d x ≤ ∑ n = N ∞ f ( n ) ≤ f ( N ) + ∫ N ∞ f ( x ) d x {displaystyle int _{N}^{infty }f(x),dxleq sum _{n=N}^{infty }f(n)leq f(N)+int _{N}^{infty }f(x),dx}     (1) ∫ n n + 1 f ( x ) d x ≤ ∫ n n + 1 f ( n ) d x = f ( n ) {displaystyle int _{n}^{n+1}f(x),dxleq int _{n}^{n+1}f(n),dx=f(n)}     (2) f ( n ) = ∫ n − 1 n f ( n ) d x ≤ ∫ n − 1 n f ( x ) d x . {displaystyle f(n)=int _{n-1}^{n}f(n),dxleq int _{n-1}^{n}f(x),dx.}     (3) ∑ n = N k ∞ 1 n ln ⁡ ( n ) ln 2 ⁡ ( n ) ⋯ ln k − 1 ⁡ ( n ) ln k ⁡ ( n ) {displaystyle sum _{n=N_{k}}^{infty }{frac {1}{nln(n)ln _{2}(n)cdots ln _{k-1}(n)ln _{k}(n)}}}     (4) ∑ n = N k ∞ 1 n ln ⁡ ( n ) ln 2 ⁡ ( n ) ⋯ ln k − 1 ⁡ ( n ) ( ln k ⁡ ( n ) ) 1 + ε {displaystyle sum _{n=N_{k}}^{infty }{frac {1}{nln(n)ln _{2}(n)cdots ln _{k-1}(n)(ln _{k}(n))^{1+varepsilon }}}}     (5) In mathematics, the integral test for convergence is a method used to test infinite series of non-negative terms for convergence. It was developed by Colin Maclaurin and Augustin-Louis Cauchy and is sometimes known as the Maclaurin–Cauchy test. Consider an integer N and a non-negative function f defined on the unbounded interval [N, ∞), on which it is monotone decreasing. Then the infinite series converges to a real number if and only if the improper integral is finite. In other words, if the integral diverges, then the series diverges as well. If the improper integral is finite, then the proof also gives the lower and upper bounds