Question

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

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Answer 1

★ 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