Question

Derive Eulers expansion of pi^2 /6​

Answer 1

Answer:

basel problem

Step-by-step explanation:

10+10=20

Answer 2

Answer:

Euler observed that the function

sin x

has roots at x = 0, ±π, ±2π, ±3π, . . .

Next, he observed that the infinite product

x

1 −

x

2

(π)

2

1 −

x

2

(2π)

2

1 −

x

2

(3π)

2

· · ·

also has roots at x = 0, ±π, ±2π, ±3π, . . .

Euler believed that these two functions are equivalent

By Maclaurin series on sin x, we find that the coefficient of the x

3

term = −1/6

On the other hand, for the infinite product, the coefficient of the x

3

term = −I/π2 = −

P∞

k=1 1/(k

2π

2

)

Thus, Euler concluded that I = π

2/6

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Step-by-step explanation: