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If 1/1^2+1/2^2+1/3^2+...oo=pi^2/6 then v...

If `1/1^2+1/2^2+1/3^2+...oo=pi^2/6` then value of `1-1/2^2+1/3^2-1/4^2+...oo=`

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(i) `(1)/(1^(2))+(1)/(2^(2)) +(1)/(3^(2) )+ ........+ infty =(pi^(2))/( 6)" "....(i) `
` therefore " "(1)/(1^(2)) +(1)/( 3 ^(2)) +(1)/( 5^(2)) +....+ infty `
` " " = ((1)/(1^(2))+ (1)/(2^(2))+(1)/(3^(2)) +(1)/(4^(2)) + ....+ infty ) -( (1)/(2^(2))+ ( 1) /(4^(2)) +(1)/( 6^(2)) + .......+ infty ) `
` =(pi^(2) )/(6) -(1)/(4) (pi^(2))/( 6) =(3)/(4) xx (pi^(2))/( 6) =(pi^(2))/( 8) `
` (ii) 1-( 1)/(2^(2)) +(1)/(3^(2) )-(1)/( 4^(2) )+ ....+ infty =(( 1)/(1^(2) ) -(1)/(3^(3) ) + .....+ -(1)/(2^(2)) -(1)/(2^(2)) + .....infty ) `
` " " =( pi^(2))/(8) -(1)/(4) xx ( pi ^(2))/( 6) =( pi ^(2))/( 12)`.
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ARIHANT MATHS-SEQUENCES AND SERIES-Exercise (Subjective Type Questions)
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  2. If the first four terms of an arithmetic sequence are a,2a,b and (a-6-...

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  3. If 1/1^2+1/2^2+1/3^2+...oo=pi^2/6 then value of 1-1/2^2+1/3^2-1/4^2+.....

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  4. If the arithmetic mean of a(1),a(2),a(3),"........"a(n) is a and b(1),...

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  6. If t(1)=1,t(r )-t( r-1)=2^(r-1),r ge 2, find sum(r=1)^(n)t(r ).

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  7. Prove that I(1),I(2),I(3)"..." form an AP, if (i) I(n)=int(0)^(pi)(...

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  8. Consider the sequence S=7+13+21+31+".....+"T(n) , find the value of T(...

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  9. Find value of (x+(1)/(x))^(3)+(x^(2)+(1)/(x^(2)))^(3)+"........"+(x^(n...

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  10. If a(m) be the mth term of an AP, show that a(1)^(2)-a(2)^(2)+a(3)^(2)...

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  11. If three unequel numbers are in HP and their squares are in AP, show ...

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  12. Ifa(1),a(2),a(3),"........",a(n) are in AP with a(1)=0, prove that (a(...

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  13. Balls are arranged in rows to form an equilateral triangle. The first ...

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  14. If theta(1),theta(2),theta(3),".......",theta(n) are in AP whose commo...

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  15. Show that,(1+5^(-1))(1+5^(-2))(1+5^(-4))(1+5^(-8))"....."(1+5^(-2n))=(...

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  16. Evaluate S=sum(n=0)^n(2^n)/((a^(2^n)+1) (where a>1).

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  17. Find the sum to infinite terms of the series tan^(-1)((1)/(3))+tan^(-1...

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  18. Find the sum to n terms, whose nth term is tan[alpha+(n-1)beta]tan (al...

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  19. If sum(r=1)^(n)T(r)=(n)/(8)(n+1)(n+2)(n+3)," find "sum(r=1)^(n)(1)/(T(...

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  20. If s1,s2,s3 denote the sum of n terms of 3 arithmetic series whose fir...

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