# Interesting Thing about Infinite Series...

$$\displaystyle\sum_{n=0}^{\infty} \frac{e}{3((n + 1)(n + 3))}$$ $$=$$ $$\frac{e}{4}$$

$\displaystyle\sum_{n=0}^{\infty} \frac{\pi}{3((n + 1)(n + 3))}$ $=$ $\frac{\pi}{4}$

$\displaystyle\sum_{n=0}^{\infty} \frac{9}{3((n + 1)(n + 3))}$ $=$ $\frac{9}{4}$

Therefore:

$\displaystyle\sum_{n=0}^{\infty} \frac{x}{3((n + 1)(n + 3))}$ will always equal $\frac{x}{4}$, where $x$ is any type of number.

$\displaystyle\sum_{n=0}^{\infty} \frac{x}{(n + 1)(n + 3)}$ will always equal $\frac{3x}{4}$, where $x$ is any type of number.

Hope you found this interesting!

# Algebra

Note by A Former Brilliant Member
10 months, 1 week ago

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- 10 months, 1 week ago

You were right! You do know more people. XD. But I don't know a thing about Calculus, Limits or Integrals, so I don't understand the note. =D

- 9 months, 3 weeks ago

$\sum_{n=0}^{\infty}\frac{1}{\left(n+1\right)\left(n+3\right)}\ =\ \frac{1}{2}\sum_{n=0}^{\infty}\left(\frac{1}{n+1}-\frac{1}{n+3}\right)=\frac{3}{4}$

So, $\sum_{n=0}^{\infty}\frac{x}{\left(n+1\right)\left(n+3\right)}$ will always be equal to $\frac{3x}{4}$.

Also, a typo in the last summation, it should be $\frac{3x}{4}$, not $\frac{x}{12}$

- 10 months, 1 week ago

Ok, I will edit it. @Aaghaz Mahajan

- 10 months, 1 week ago

Nice post @Yajat Shamji. Thanku for sharing this with us.

Alternatively, you could go from your result to observation and proof it like @Aaghaz Mahajan mentioned . Found your post interesting.

- 10 months, 1 week ago

Hey man, are you the younger brother of Aniket Sanghi?? i used to follow his feed for those amazing problems....no doubt he secured a great rank in JEE!!!

- 10 months, 1 week ago

Thanks, @Aryan Sanghi! But I am GCSE so am only posting this for comments and suggestions

- 10 months, 1 week ago

Can you please send a proof ? Also you can prove more sums like this via the Digamma function or atleast that's how I solved your sums.Try Brilliant website for the Digamma function.

- 10 months, 1 week ago

Look at my post to Aryan Sanghi.

- 10 months, 1 week ago

Knowing the convergence of the series, this result is trivial if the sum can be calculated.

- 2 months, 3 weeks ago