Math, asked by Madhu123mri, 1 year ago


 {1}^{2}  +  {2}^{2} +   {3}^{2}  +  {4 }^{2}  +  {5}^{2} ..... \: upto \: infinity
= ??????

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Answers

Answered by δΙΔΔΗλΣΓΗΛ
17
★ INFINITY ★

1 {}^{2}  + 2 {}^{2}  + 3 {}^{2}  + ... \infty  \\
Lets initiate for the ' nth ' value of series -

Sum of squares of first n natural numbers =
 \frac{n(n + 1)(2n + 1)}{6}
Now consider the properties of INFINITY -

☣ ∞ ± C = ∞ , ∀ C ∈ R
☣ - ∞ ± C = - ∞ ∀ C ∈ R

 {c}^{ \infty }  =  \infty  \\ c > 1 \\  = 0 \: if \: 0 \leqslant c < 1 \\  = 1 \: if \: c = 1

☣ ∞ ( ∞ ) = ∞

Now either n → ∞ ( n tends to INFINITY )
or n = ∞

Non of the above operations goes IN INDETERMINATE FORM

( \infty  + 1) =  \infty  \\ (2 \infty ) =  \infty  \\ (2 \infty  + 1) = ( \infty  + 1) =  \infty  \\ ( \infty )( \infty  + 1)(2 \infty  + 1) = ( \infty )( \infty )( \infty ) \\  { \infty }^{ \infty }  =  \infty  \\ by \: using \: this \: property \\  { \infty }^{3}  =  \infty  \\ simultaneously
Now
 \frac{ \infty }{6}  =  \infty
Is the converse of above property simultaneously
So ,
( \infty ) \times (c) =  \infty  \\ ( \infty ) \times (6) =  \infty
Hence , the final system reduces to

INFINITY
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