Question

1+1/2²+1/3²+......+1/n²=?

Answer 1

_Monotonia

an=1+1/2²+1/3²+...+1/n²

an+1=1+1/2²+1/3³+...+1/n²+1/(n+1)²

an+1-an=1/(n+1)²>0=>

an+1>an=>

an   sir monoton  crescator

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1≤1

1/2²<1/1*2=1-1/2

1/3²<1/2*3=1/2-1/3

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1/n²<1/n-1/(n+1)=1/(n-1)-1/(n+1)

--------------------------Se   aduna

an=∑1/k²=1+1-1/2+1/2-1/3+...+1/(n-1)-1/(n+1)=

2-1/(n+1)=(2n+2-1)/n+1)=(2n+1)/(n+1)→2

Aplici  teorema,  daca  un  sir  e   marginit si   e   crescator  ,   atunci  limita   sa    este  marginea   sa   superioara.In  cazul   de  fata  2.

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