InterviewSolution
Saved Bookmarks
InterviewSolution
| 1. |
Find the sum of the series `1+2^(2)x+3^(2)x^(2)+4^(2)x^(3)+"...."" upto "infty|x|lt1`. |
|
Answer» Here, the numbers `1^(2),2^(2),3^(2),4^(2),"...."i.e.1,4,9,16"...."` are not in AP but ` 1,4-1=3,9-4=5,16-9=7,"...."` are in AP. Let `S_(infty)=1+2^(2)x+3^(2)x^(2)+4^(2)x^(3)+"...."" upto "infty` `=1+4x+9x^(3)+16x^(3)+"...."" upto "infty "......(i)"` Multiplying both sides of Eq. (i) by x, we get `S_(infty)=x+4x^(2)+9x^(3)+16x^(4)+"...."" upto "infty "......(ii)"` Subtracting Eq.(ii) from Eq.(i), we get `(1-x)S_(infty)=1+3x+5x^(2)+7x^(3)+"...."" upto "infty "......(iii)"` Again, multiplying both sides of Eq. (iii) by x, we get `x(1-x)S_(infty)=x+3x^(2)+5x^(3)+7x^(4)+"...."" upto "infty "......(iv)"` Subtracting Eq. (iv) from Eq. (iii),we get `(1-x)(1-x)S_(infty)=1+2x+2x^(2)+2x^(3)+"...."" upto "infty` `=1+2(x+x^(2)+x^(3)+"...."" upto "infty)` `=1+2((x)/(1-x))=((1+x)/(1-x))` ` therefore S_(infty)=((1+x))/((1-x)^(3))` |
|