Section 11.1

The Definite Integral as the Limit of a Sum

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Problem 4541

Using definite integrals, find the limits of the following sums:

lim⁡n→∞(1n+2n+3n+…+2n−1n)⋅1n. \lim _{n \rightarrow \infty } \left(\frac{1}{n}+\frac{2}{n}+\frac{3}{n}+\ldots +\frac{2n-1}{n}\right) \cdot \frac{1}{n} .
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Problem 4542

Using definite integrals, find the limits of the following sums:

lim⁡n→∞(13n4+23n4+33n4+…+(4n−1)3n4). \lim _{n \rightarrow \infty } \left(\frac{1^{3}}{n^{4}}+\frac{2^{3}}{n^{4}}+\frac{3^{3}}{n^{4}}+\ldots +\frac{(4n-1)^{3}}{n^{4}}\right) .
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Problem 4543

Using definite integrals, find the limits of the following sums:

lim⁡n→∞πn(sin⁡πan+2πan+…+(n−1)πan). \lim _{n \rightarrow \infty } \frac{\pi }{n} \left(\sin \frac{\pi a}{n}+\frac{2\pi a}{n}+\ldots +\frac{(n-1)\pi a}{n}\right) .
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Problem 4544

Using definite integrals, find the limits of the following sums:

lim⁡n→∞(1n+1+1n+2+…+12n). \lim _{n \rightarrow \infty } \left(\frac{1}{n+1}+\frac{1}{n+2}+\ldots +\frac{1}{2n}\right) .
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Problem 4545

Using definite integrals, find the limits of the following sums:

lim⁡n→∞(nn2+12+nn2+22+…+nn2+(n−1)2). \lim _{n \rightarrow \infty } \left(\frac{n}{n^{2}+1^{2}}+\frac{n}{n^{2}+2^{2}}+\ldots +\frac{n}{n^{2}+(n-1)^{2}}\right) .
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Problem 4546

Using definite integrals, find the limits of the following sums:

lim⁡n→∞(24n2−12+24n2−22+…+24n2−n2). \lim _{n \rightarrow \infty } \left(\frac{2}{\sqrt{4n^{2}-1^{2}}}+\frac{2}{\sqrt{4n^{2}-2^{2}}}+\ldots +\frac{2}{\sqrt{4n^{2}-n^{2}}}\right) .
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Problem 4547

Find

lim⁡n→∞(n+1)(n+2)…2nnn. \lim _{n \rightarrow \infty } \frac{\sqrt[n]{(n+1)(n+2)\ldots 2n}}{n} .
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Problem 4548

Find the value of the integral ∫01ex dx\int_{0}^{1} e^{x}\, dx by regarding it as the limit of a sum.

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Problem 4549

Find, for b>a>0b > a > 0, the value of the integral ∫abxn dx\int_{a}^{b} x^{n}\, dx by regarding it as the limit of the sum ∑xνnΔxν\sum x_{\nu }^{n} \Delta x_{\nu }, where the values xνx_{\nu } form a geometric progression. (Fermat.)

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Problem 4550

Starting from the partial-fraction decomposition:

nxn−1xn−1=1x−x0+1x−x1+…+1x−xn−1,xν=e2πνin, \frac{n x^{n-1}}{x^{n}-1} = \frac{1}{x-x_{0}}+\frac{1}{x-x_{1}}+\ldots +\frac{1}{x-x_{n-1}}, \quad x_{\nu } = e^{\frac{2\pi \nu i}{n}},

obtain the equality:

∫02πdφx−eφi={0,for ∣x∣<1;2πx,for ∣x∣>1. \int _{0}^{2\pi } \frac{d\varphi }{x-e^{\varphi i}} = \begin{cases} 0, & \text{for } \left|x\right| < 1; \\ \dfrac {2\pi }{x}, & \text{for } \left|x\right| > 1 . \end{cases}
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Problem 4551

Starting from the formula

x2n−1=(x2−1)∏k=1n−1(x2−2xcos⁡kπn+1), x^{2n}-1 = (x^{2}-1) \prod _{k=1}^{n-1} \left(x^{2}-2x\cos \frac{k\pi }{n}+1\right),

obtain the value of the Poisson integral:

∫0πln⁡(1−2xcos⁡φ+x2) dφ={0,for ∣x∣<1;πln⁡(x2),for ∣x∣>1. \int _{0}^{\pi } \ln (1-2x\cos \varphi +x^{2})\, d\varphi = \begin{cases} 0, & \text{for } \left|x\right| < 1; \\ \pi \ln (x^{2}), & \text{for } \left|x\right| > 1 . \end{cases}
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Problem 4552

From the partial-fraction decomposition:

2nx2n−1x2n−1=1x2−1+∑k=1n−12x−2cos⁡kπnx2−2xcos⁡kπn+1 \frac{2n x^{2n-1}}{x^{2n}-1} = \frac{1}{x^{2}-1}+\sum _{k=1}^{n-1} \frac{2x-2\cos \frac{k\pi }{n}}{x^{2}-2x\cos \frac{k\pi }{n}+1}

obtain the equality:

∫0πx−cos⁡φx2−2xcos⁡φ+1 dφ={0,for ∣x∣<1;πx,for ∣x∣>1. \int _{0}^{\pi } \frac{x-\cos \varphi }{x^{2}-2x\cos \varphi +1}\, d\varphi = \begin{cases} 0, & \text{for } \left|x\right| < 1; \\ \dfrac {\pi }{x}, & \text{for } \left|x\right| > 1 . \end{cases}
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Problem 4553

The function [u]\left[u\right] equals the integer part of the variable uu, i.e. the integer mm such that m⩽u<m+1m \leqslant u < m+1. In that case

[2u]−2[u]={0,if u−m<12;1,if 12⩽u−m<1. \left[2u\right]-2\left[u\right] = \begin{cases} 0, & \text{if } u-m < \dfrac {1}{2}; \\ 1, & \text{if } \dfrac {1}{2} \leqslant u-m < 1 . \end{cases}

Prove that for the function f(x)=[2x]−2[1x]f(x) = \left[\dfrac {2}{x}\right]-2\left[\dfrac {1}{x}\right] on the interval (0,1)(0,1) the following inequality holds:

∑ν=1nOνΔxν<3max⁡Δxν, \sum _{\nu =1}^{n} O_{\nu }\Delta x_{\nu } < 3\sqrt{\max \Delta x_{\nu }},

and hence the function f(x)f(x) is Riemann integrable.

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Problem 4554

The function f(x)f(x) is defined by:

{f(x)=0,if x is irrational;f(x)=1q,if x=pq, \begin{cases} f(x) = 0, & \text{if } x \text{ is irrational}; \\ f(x) = \dfrac {1}{q}, & \text{if } x = \dfrac {p}{q}, \end{cases}

where pp and qq are relatively prime integers. Prove that on the interval (0,1)(0,1) the following inequality holds:

∑ν=1nOνΔxν<2max⁡Δxν. \sum _{\nu =1}^{n} O_{\nu }\Delta x_{\nu } < 2\sqrt{\max \Delta x_{\nu }} .

Thus f(x)f(x) is integrable on the interval (0,1)(0,1), although it has a discontinuity at every point with rational abscissa.

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Problem 4555

The function f(x)f(x) is defined by f(x)=0f(x) = 0 for xx irrational and f(x)=1f(x) = 1 for xx rational. Prove that it is not integrable on any interval.

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