Integral criterion

Lower estimate of the harmonic series by area under the function 1 / x

The integral criterion (also integral comparison criterion ) is a mathematical convergence criterion for infinite series . The row is viewed as an area under a step function , which is estimated by the area under a curve . Convergence can be demonstrated with an upward estimate , and divergence downward. The area under the curve is calculated using the integral .

formulation

Let it be a monotonically decreasing function which is defined on the interval with an integer and only takes on nonnegative values. Then the series converges if and only if the integral exists, that is, if it takes on a finite value. Instead of the existence of the integral one sometimes also speaks - synonymously - of the convergence of the integral. ${\displaystyle f}$ ${\displaystyle [p,\infty )}$ ${\displaystyle p}$ ${\displaystyle \textstyle \sum _{n=p}^{\infty }f(n)}$ ${\displaystyle \textstyle \int _{p}^{\infty }f(x)\,\mathrm {d} x}$

More precisely: If it is monotonically decreasing, then applies ${\displaystyle p\in \mathbb {Z} ,f\colon [p,\infty )\to [0,\infty )}$

${\displaystyle f}$is integrable is convergent. ${\displaystyle [p,\infty )}$ ${\displaystyle \iff \sum _{n=p}^{\infty }f(n)}$

If one of the two, i.e. existence of the integral or convergence of the series, and thus also the other, applies, the estimates apply

${\displaystyle \sum _{n=p+1}^{\infty }f(n)\leq \int _{p}^{\infty }f(x)\,\mathrm {d} x\leq \sum _{n=p}^{\infty }f(n)}$.

Examples

To see if the series

${\displaystyle \sum _{n=1}^{\infty }{\frac {1}{n^{2}}}}$

converges, it is found that it is with the function

${\displaystyle f\colon [1,\infty )\to \mathbb {R} ,\quad f(x)={\frac {1}{x^{2}}}}$

than can be written. The function is monotonically decreasing in the interval and the following applies: ${\displaystyle \sum _{n=1}^{\infty }f(n)}$ ${\displaystyle f}$ ${\displaystyle I=[1,\infty )}$

${\displaystyle \int \limits _{1}^{\infty }{\frac {1}{x^{2}}}\,\mathrm {d} x=\lim _{b\to \infty }\left[-{\frac {1}{x}}\right]_{1}^{b}=1.}$

The integral is finite and according to the integral criterion the series is convergent.

Similarly, can harmonic series with than to be rewritten. The function decreases monotonically in the interval , which means that the integral criterion can be applied: ${\displaystyle \sum _{n=1}^{\infty }{\frac {1}{n}}}$ ${\displaystyle f\colon [1,\infty )\to \mathbb {R} ,\quad f(x)={\frac {1}{x}}}$ ${\displaystyle \sum _{n=1}^{\infty }f(n)}$ ${\displaystyle f}$ ${\displaystyle I=[1,\infty )}$

${\displaystyle \int _{1}^{\infty }{\frac {1}{x}}\,\mathrm {d} x=\lim _{b\to \infty }{\bigg [}\ln(x){\bigg ]}_{1}^{b}=+\infty }$

The integral is divergent and so is the harmonic series.

illustration

The integral criterion is accessible just by looking at it: The last line in particular resembles a popular justification of the concept of the Riemann integral with the help of upper and lower sums.

Because according to the assumption it falls monotonically, the largest and the smallest function value are on each interval (with an integer ) on this interval. Because the interval has a width of 1, the area under is always less than or equal to and greater than or equal to . If the integral or the series converges, the other expression must also converge. ${\displaystyle f}$ ${\displaystyle [q,q+1]}$ ${\displaystyle q}$ ${\displaystyle f(q)}$ ${\displaystyle f(q+1)}$ ${\displaystyle f}$ ${\displaystyle f(q)\cdot 1}$ ${\displaystyle f(q+1)\cdot 1}$

Or: The series converges, i.e. approaches the limit value from infinitely close. For the integral, this means that the area no longer increases, but also approaches a (area) value. If the area had no limit towards infinity, a value could never be fixed for the integral and thus the integral would not have a finite value, which contradicts the above definition. ${\displaystyle \textstyle \sum _{n=p}^{\infty }f(n)}$ ${\displaystyle p}$ ${\displaystyle \textstyle \int _{p}^{\infty }f(x)\mathrm {d} x}$

literature

• Konrad Königsberger : Analysis 1 . Springer, Berlin 2004, ISBN 3-540-41282-4