Is there a closed formula for the sum: ∑ n k = 1 1 k 2 ∑ 𝑘 = 1 𝑛 1 𝑘 2 ?

byOnline job offer WEBSITE [All] -May 25, 2024

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To find a closed formula for the sum

\sum_{𝑘 = 1}^{𝑛} \frac{1}{𝑘^{2}}

we first recognize that this sum is a well-known series in mathematics, specifically the sum of the reciprocals of the squares of the first $𝑛$ natural numbers. The infinite series for this sum is known and is given by the Basel problem result:

\sum_{𝑘 = 1}^{\infty} \frac{1}{𝑘^{2}} = \frac{𝜋^{2}}{6} .

However, for a finite $𝑛$ , there isn't a simple closed formula for $\sum_{𝑘 = 1}^{𝑛} \frac{1}{𝑘^{2}}$ in terms of elementary functions. Nevertheless, we can express it in terms of the digamma function $𝜓 (𝑥)$ , which is the derivative of the logarithm of the gamma function $Γ (𝑥)$ , and it has a relation with the Hurwitz zeta function $𝜁 (𝑠, 𝑞)$ . Specifically,

\sum_{𝑘 = 1}^{𝑛} \frac{1}{𝑘^{2}} = 𝜁 (2) - 𝜁 (2, 𝑛 + 1),

where $𝜁 (𝑠, 𝑞)$ is the Hurwitz zeta function.

For practical purposes, the sum $\sum_{𝑘 = 1}^{𝑛} \frac{1}{𝑘^{2}}$ can be approximated quite well using numerical methods, or we can use partial sums and asymptotic expansions for large $𝑛$ .

To summarize, there isn't a simple closed form like a polynomial or elementary function, but the sum $\sum_{𝑘 = 1}^{𝑛} \frac{1}{𝑘^{2}}$ is well-understood and can be computed or approximated using special functions and numerical methods.

Is there a closed formula for the sum: ∑ n k = 1 1 k 2 ∑ 𝑘 = 1 𝑛 1 𝑘 2 ?

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