Good question.

Here is an interesting way to think about it.

\sum_{n=1}^{\infty} \frac{1}{n^2} = \int_1^{\infty} \frac{1}{\lfloor x \rfloor^2} dx .

\int_1^{\infty} \frac{1}{x^2} dx is the area under \frac{1}{x^2} in [math] [1, \infty) [/math], whereas \sum_{n=1}^{\infty} \frac{1}{n^2} is the area under \frac{1}{\lfloor x \rfloor^2} in [math] [1, \infty) [/math].

This is illustrated in the following picture (where the green curve is 1/x^2 , the solid red curve is 1/n^2 and the dotted red curve is 1/(n+1)^2 .

One can see from the picture that \int_1^{\infty} \frac{1}{x^2} dx < \sum_{n=2}^\infty \frac{1}{n^2} + 1/2 (\sum_{n=1}^\infty \frac{1}{n^2} - \sum_{n=2}^\infty \frac{1}{n^2}) = \sum_{n=1}^{\infty} \frac{1}{n^2} - 1/2 \approx 1.145

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calculus - Nice proofs of $\zeta(4) = \frac{\**pi**^4}{90}$? - Mathematics ...

I know some nice ways to prove that ζ(**2**)=∑∞n=**11n2**=**π2/6**. ... If we substitute **π**
for **x** in the Fourier trigonometric series expansion of f(**x**)=**x**4, with ..... **It** is **not** hard
to guess that p(z) is **same** as isin(z)×sin(zi)z**2**=(**1**−z**2**3!+z45! .... Consider the
contour **integral** ∮C**π**cot**π**zz4 dz ..... The second sum in the left member is **equal**
to:.

For more information, see calculus - Nice proofs of $\zeta(4) = \frac{\**pi**^4}{90}$? - Mathematics ...

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**Pi** Formulas -- from Wolfram MathWorld

**pi** is intimately related to the properties of circles **and** spheres. For a circle of
radius r ... terms is **approx** (3/4)^k . An **infinite** sum series to Abraham Sharp (ca.
1717) is ... 105-106). The coefficients can be found from the **integral** ... At the cost
of a square root, Gosper has noted that **x**=**1**/**2** gives **2** ... 1999) **and** is **equivalent**
to ...

For more information, see **Pi** Formulas -- from Wolfram MathWorld

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Gamma Function -- from Wolfram MathWorld

8). **It** is analytic everywhere except at z=0 , -**1** , -**2** , ..., **and** the residue at z=-k is ...
The gamma function is implemented **in** the Wolfram Language as Gamma[z]. ...
**Integrating** equation (3) by parts for a real argument, **it** can be seen that ... The
gamma function can also be defined by an **infinite** product form (Weierstrass form
) ...

For more information, see Gamma Function -- from Wolfram MathWorld

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calculus - Evaluating the **integral** $\int_0^\infty \frac{\sin **x**} **x** \ **dx** ...

**Integration** by parts yields Dn=**12**n+**1**∫**π**/**2**0f′(x)cos(**2**n+**1**)**x** dx=O(**1**/**n**). .... For θ
**not** an integer multiple of **2π**, we have ∑einθn=−log(**1**−eiθ). .... Here is a sketch
of another elementary solution based on a proof in Bromwich's Theory of **Infinite**
Series. ...... to deduce **it** is exactly **1**sinx, so the RHS of (**1**) simply **equals π**.

For more information, see calculus - Evaluating the **integral** $\int_0^\infty \frac{\sin **x**} **x** \ **dx** ...

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differential equations - What is wrong with my approach to solving a ...

Jan 18, 2019 **...** **Infinity** -> 20 // Activate, **x**] // Evaluate, {**x**, 0, **Pi**}, {t, 0, **1**}] ..... your Exp[-**n**^**2** t] only
damps out part of the sum **while** the sum damps out very rapidly **in** the separate
variable case. .... Again, each side must be **equal** to a constant. .... c3***Integrate**[
Cos[**n*****x**]^**2**, {**x**, 0, **Pi**}] == (**1**/**Pi**)***Integrate**[**x**^**2***Cos[**n*****x**], {**x**, 0, **Pi**}] ...

For more information, see differential equations - What is wrong with my approach to solving a ...

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Basel problem - Wikipedia

Squaring the circle; Basel problem; Six nines in **π** · Other topics related to **π** · v · t ·
e. The Basel problem is a problem in mathematical analysis with relevance to
number theory, first posed by Pietro Mengoli in 1650 and solved by Leonhard
Euler in 1734 **and** read on 5 December 1735 in The ... The sum of the series is
**approximately equal** to 1.644934.

For more information, see Basel problem - Wikipedia

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Harmonic series (mathematics) - Wikipedia

History[edit]. The fact that the harmonic series diverges was first proven **in** the
14th century by Nicole Oresme, but this achievement fell into obscurity. Proofs
were given **in** the 17th century by Pietro Mengoli, Johann Bernoulli, ... If the worm
travels **1** centimeter per minute **and** the band stretches **1** meter per minute, will
the ...

For more information, see Harmonic series (mathematics) - Wikipedia