Pies all over the floor!

Hello everybody! This is my very first note. As the title suggests, today we are going to talk about something related to pi. Nah, not the chicken pie or apple pie or any other pies that you enjoy eating but the pi defined as the ratio of circumference to the diameter of a circle!

Well, to get things started, find π+π2+π3+...+π10\lfloor \pi\rfloor+\lfloor \pi^{2}\rfloor+\lfloor \pi^{3}\rfloor+...+\lfloor \pi^{10}\rfloor without using a calculator. Actually, this is a question which I propose yet fail to solve until now.

More generally, find π+π2+π3+...+πn\lfloor \pi\rfloor+\lfloor \pi^{2}\rfloor+\lfloor \pi^{3}\rfloor+...+\lfloor \pi^{n}\rfloor in terms of nn.

I hope someone can offer help or suggestions. If you find this interesting just like I do, help me like or re share this note to attract the attention of pros, thank you. I'll sign off here for now.

Cheers!

Note by Donglin Loo
1 year, 3 months ago

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1 vote

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π=3.14    π=3\pi = 3.14 \implies \lfloor \pi\rfloor= 3

So, we have to find π+π2+π3+...+π10\lfloor \pi\rfloor+\lfloor \pi^{2}\rfloor+\lfloor \pi^{3}\rfloor+...+\lfloor \pi^{10}\rfloor

    3+32+33+34+35....+310\implies 3 + 3^2 + 3^3 + 3^4 + 3^5 .... + 3^{10}

The above series is in G.P with a=3,r=3,n=10a = 3, r = 3, n = 10

Sum of terms in G.P = 3(3101)31\dfrac{3(3^{10} - 1)}{3 - 1}

    1.5(3101)\implies 1.5(3^{10} - 1)

Coming to the next part of the question sum of infinite terms in an increasing G.P is infinite.

However, if nn is finite the formula would be :

3(3n1)31\dfrac{3(3^n -1)}{3 - 1}

Ram Mohith - 1 year, 3 months ago

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@Ram Mohith. You can't assume π=3\pi=3 just like that. For instance, π3=31.000627668\pi^3=31.000627668 so π3=31\lfloor \pi^3\rfloor=31 instead of 33=273^3=27. You can verify this with a calculator. The difficult part is that π=3.14159265\pi=3.14159265 and with exponentiation things can get very random.

donglin loo - 1 year, 3 months ago

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Interesting question, may use this identity 1+14+19+116+=π261+\dfrac14+\dfrac19+\dfrac1{16}+\cdots=\dfrac{\pi^2}6 to do that but I'm not sure...

Kelvin Hong - 1 year, 3 months ago

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Oh wow. I did not think of this. Thanks for your suggestion.

donglin loo - 1 year, 3 months ago

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Take a look at this OEIS ,though it may not be helpful.

X X - 1 year, 3 months ago

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Oh I see. Thanks for your idea

donglin loo - 1 year, 3 months ago

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Perhaps utilize π2=2244668813355779\dfrac{\pi}{2}=\dfrac{2\cdot2\cdot4\cdot4\cdot6\cdot6\cdot8\cdot8\cdots}{1\cdot3\cdot3\cdot5\cdot5\cdot7\cdot7\cdot9\cdots}?

Kenneth Tan - 1 year, 3 months ago

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Wow. Is there a proof for this?

donglin loo - 1 year, 3 months ago

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Wallis Product

X X - 1 year, 3 months ago

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@X X Thanks @X X

donglin loo - 1 year, 3 months ago

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