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Evaluate : sum(r=1)^oo r^2x^(r-1),(|x|<1...

Evaluate : `sum_(r=1)^oo r^2x^(r-1),(|x|<1)`.

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A DAS GUPTA-Progression, Related Inequalities and Series-Exercise
  1. 1-4/2+7/(2^2)-10/(2^3)+

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  2. Sum to infinite terms : 1*2+2*3x+3*4*x^2+...,(|x|<1).

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  3. Evaluate : sum(r=1)^oo r^2x^(r-1),(|x|<1).

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  4. Evaluate : sum(n=1)^(15) (2n+1)2^n.

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  5. If f(x) = x+3x^2+5x^4+7x^8+... to n terms then find the value of f'(1)...

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  6. If g(x)=1+3x+5x^2+7x^3+... to n terms then find int0^1 g(x)dx.

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  7. Show that (n-1)/(n+1)+3((n-1)/(n+1))^2+5((n-1)/(n+1))^3+....+oo=sum(r=...

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  8. Find the sum to n terms of the series 1*2^2 + 2*3^2 + 3*4^2 + ...

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  9. (x^2+1/(x^2)+2)+(x^4+1/(x^4)+5)+(x^6+1/(x^6)+8)+

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  10. Sum to infinite terms : 1/5-2/7^2+3/5^3-4/7^4+....

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  11. Sum to n terms : 0.4+0.44+0.444+....

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  12. Find the sum to n terms of the series 5 + 55 + 555 + ...

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  13. Natural numbers are divided into groups in the following way: 1,(2,3),...

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  14. Prove that the sum of the numbers in any group in the following is the...

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  15. Find the sum of the n terms of the sequence 1/(1+1^2+1^4)+2/(1+2^2+2^...

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  16. Sum to n terms of the series (1)/((1+x)(1+2x))+(1)/((1+2x)(1+3x))+(1)/...

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  17. The sum of the series : 1/(1*2*3*4)+1/(3*4*5*6)+1/(5*6*7*8)+...+oo is

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  18. Sum to n terms of the series 1/(5!)+1/(6!)+(2!)/(7!)+(3!)/(8!)+.........

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  19. ((1)/(x+1)+(1)/(x+5))=((1)/(x+2)+(1)/(x+4)):

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  20. Evaluate sum(n=1)^n (n^2+n-1)/((n+2)!).

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