Infinite sums and products 1 (#9)

Calculus Level 2

$\large 1 + \frac {1}{9} + \frac {1}{25} + \frac {1}{49} + \frac {1}{81} + \ldots$

Find the sum above to two decimal places.

The answer is 1.23.

2 solutions

Pranjal Jain
Dec 26, 2014

We will be using the euler's solution to Basel problem whose proof can be found here .

The result (worth remembering) is $\displaystyle\sum_{n=1}^{\infty} \dfrac{1}{n^{2}}=\dfrac{\pi^{2}}{6}$

Let $S'=1+\dfrac{1}{9}+\dfrac{1}{25}+\dfrac{1}{49}+...$

$S=1+\dfrac{1}{4}+\dfrac{1}{9}+\dfrac{1}{16}+\dfrac{1}{25}+\dfrac{1}{36}...\\=(1+\dfrac{1}{9}+\dfrac{1}{25}+...)+(\dfrac{1}{4}+\dfrac{1}{16}+\dfrac{1}{36}+...)\\=S'+\dfrac{1}{4}(1+\dfrac{1}{4}+\dfrac{1}{9}+...)\\=S'+\dfrac{S}{4}\\\Rightarrow\dfrac{3S}{4}=S'=\dfrac{3}{4}×\dfrac{\pi^{2}}{6}=\dfrac{\pi^{2}}{8}$

Great method solving without Zeta function!

찬홍 민 - 6 years, 5 months ago

Thanks! Should I calculate value of $\sum\dfrac{1}{n^{2}}$ directly here? Or that link would work?

Pranjal Jain - 6 years, 5 months ago

I just want to know how to calculate some simple values of gamma function. Yes that will help!

찬홍 민 - 6 years, 5 months ago

@찬홍 민 – Gamma function? Do you mean the one which is an extension of the factorial function, i.e., $\Gamma(n)=(n-1)!$ ?

Prasun Biswas - 6 years, 4 months ago

Well, if I remember correctly, $\zeta (2)= \displaystyle \sum_{i=1}^\infty \dfrac{1}{i^2}$ which is equivalent to the Basel series. So, he used the zeta function after all.

Prasun Biswas - 6 years, 4 months ago

As i dont know the answer to this problem so i m asking it here : Now if

${ S }_{ i }=\sum _{ k=1 }^{ \infty }{ \frac { i }{ { ({ 36k }^{ 2 }-1) }^{ i } } }$

Find ${ S }_{ 1 }+{ S }_{ 2 }$ ?????????????? pls help

Calvin Lin ,megh choksi ,Adarsh Kumar ,Pratik Shastri ,Ronak Agarwal ,Sandeep Bhardwaj ,Sanjeet Raria

Gautam Sharma - 6 years, 4 months ago

Brock Brown
Dec 26, 2014

from fractions import Fraction as f
total = 0
n = 1
infinity = 350 #infinity is arbitrary
while n <= infinity:
    total += f(1,n**2)
    n += 2
print total.numerator/float(total.denominator)

Infinite sums and products 1 (#9)

The answer is 1.23.

2 solutions

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