Squared harmonic series

Calculus Level 1

The harmonic series 11+12+13+14+\frac { 1 }{ 1 } +\frac { 1 }{ 2 } +\frac { 1 }{ 3 } +\frac { 1 }{ 4 }+ \cdots appears to approach a single number as the number of terms increases. However, as the number of terms approaches infinity, the sum also approaches infinity!

Based on this information, does the sum 112+122+132+142+\frac { 1 }{ { 1 }^{ 2 } } +\frac { 1 }{ { 2 }^{ 2 } } +\frac { 1 }{ { 3 }^{ 2 } } +\frac { 1 }{ { 4 }^{ 2 } } +\cdots also approach infinity as the number of terms approaches infinity?

(Challenge: Solve this problem WITHOUT using the riemann zeta function.)

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