In this paper we deal with real harmonic series, without considering their complex extension to the Riemann zeta function. It is well known that the generalized harmonic series are convergent if the exponent is greater than one, while they are divergent if the exponent is one or less than one. Further, if the exponent is an even natural number 2k, there exists the sum of the series in closed form being equal to π^{2k} times a rational number. This sum was calculated for the first time by Euler through Taylor’s expansion of the function sin x/x and then by Fourier through the expansion of suitable periodic functions. In recent times the formula of the sum of the harmonic series with exponent 2 has been proved in many other ways through elementary goniometric arguments or simple properties of the series and product expansions. Many of these methods, however, apply only to the case of exponent 2. In this paper we obtain the sum of all generalized harmonic series with an even natural exponent through an operatorial method by calculating the eigenvalues of the differential operators derivative of order 2k defined on a certain Hilbert space and then by inverting such operators, in order to obtain the sum of the series as trace of the inverse operators.

Sum of the generalized harmonic series with even natural exponent

PATRI', Stefano
2013

Abstract

In this paper we deal with real harmonic series, without considering their complex extension to the Riemann zeta function. It is well known that the generalized harmonic series are convergent if the exponent is greater than one, while they are divergent if the exponent is one or less than one. Further, if the exponent is an even natural number 2k, there exists the sum of the series in closed form being equal to π^{2k} times a rational number. This sum was calculated for the first time by Euler through Taylor’s expansion of the function sin x/x and then by Fourier through the expansion of suitable periodic functions. In recent times the formula of the sum of the harmonic series with exponent 2 has been proved in many other ways through elementary goniometric arguments or simple properties of the series and product expansions. Many of these methods, however, apply only to the case of exponent 2. In this paper we obtain the sum of all generalized harmonic series with an even natural exponent through an operatorial method by calculating the eigenvalues of the differential operators derivative of order 2k defined on a certain Hilbert space and then by inverting such operators, in order to obtain the sum of the series as trace of the inverse operators.
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/11573/504676
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