Abstract
As we saw in Theorem 7.92, the sum of the series \(\sum _{n=1}^\infty 1/n^{2k}\) can be given explicitly for every positive integer k. These series, however, are exceptions: in general, the sum of an arbitrary series cannot be expressed in closed form. In fact, no closed expression is known for the sum \(\sum _{n=1}^\infty 1/n^3\).
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Notes
- 1.
The formula was introduced by Jacob Bernoulli in his book on probability theory (1713).
- 2.
It might happen that the indefinite integral of an elementary function is not elementary, yet we can express its definite integral in closed form on some intervals. For example, the indefinite integrals of the functions \(x\cdot \mathrm{ctg}\, {x}\) and \(x^2 /\sin ^2 x\) are not elementary, but the value of their integrals on the interval \([0,\pi /2]\) are \(\pi \cdot \log 2\) and \(2\pi \cdot \log 2\), respectively. See Problems 19.20 and 19.22 in [7].
- 3.
Thomas Simpson (1710–1761), British mathematician.
- 4.
Obviously, the parameter can be denoted by any letter. In choosing the letter t (instead of the letter c) we want to indicate that we consider the parameter to be a variable, i.e., we want to think of the value of the integral as a function of t.
- 5.
The two cases do not exclude each other: if H is a bounded interval and \(f_t\) is Riemann integrable on H for every \(t\in T\), then the integral \(\int f(t, x)\, dx\) is both a parametric Riemann integral and a parametric improper integral. See [7, Remark 19.4.2].
- 6.
The function \(D_1 f\) is integrable on \([a, b]\times H\), since it is bounded and continuous there. See Theorem 4.14.
- 7.
Henri Lebesgue (1875–1941), French mathematician.
- 8.
Charles Jean de la Vallée Poussin (1866–1962), Belgian mathematician.
- 9.
Luitzen Egbertus Jan Brouwer (1881–1966), Dutch mathematician.
- 10.
Giuseppe Peano (1858–1932), Italian mathematician.
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Laczkovich, M., Sós, V.T. (2017). Miscellaneous Topics. In: Real Analysis. Undergraduate Texts in Mathematics, vol 3. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-7369-9_8
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