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If X=1+a+a^(2)+a^(3)+"..."+infty " and "...

If `X=1+a+a^(2)+a^(3)+"..."+infty " and " y=1+b+b^(2)+b^(3)+"..."+infty "` show that `1+ab+a^(2)b^(2)+a^(3)b^(3)+"..."+infty=(xy)/(x+y-1), "where " 0ltalt1" and "0ltblt1`.

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ICSE-SEQUENCE AND SERIES -EXERCISE 14 (f)
  1. Of how many terms is ,(55)/(72) the sum of the series (2)/(9) -(1)/(3...

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  2. The second term of a G.P. is 2 and the sum of infinite terms is 8. Fin...

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  3. Find the value of 0.23434343434..... regarding it as a geometric serie...

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  4. Evaluate : (a) 0.9bar7 (b) 0.2345

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  5. Find a rational number which when expressed as a decimal will have 1.2...

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  6. If a+b+.... + l is a G.P., prove that its sum is (bl-a^(2))/(b-a) .

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  7. The nth term of a geometrical progression is (2^(2n-1))/(3) for all va...

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  8. A geometrical progression of positive terms and an arithmetical progre...

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  9. In a geometric progression, the third term exceeds the second by 6 and...

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  10. In an infinite geometric progression, the sum of first two terms is 6 ...

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  11. Three numbers are in A.P. and their sum is 15. If 1,4 and 19 be added ...

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  12. Calculate the least number of terms of the geometric progression 5 + 1...

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  13. If S is the sum, P the product and R the sum of the reciprocals of n t...

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  14. Find the sum of the first n terms of the series: 0.2 + 0.22 + 0.222+...

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  15. If (2)/(3)=(x-(1)/(y))+(x^(2)-(1)/(y^(2)))+ ... "To" oo and xy =2 th...

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  16. S(1),S(2), S(3),...,S(n) are sums of n infinite geometric progressions...

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  17. Find three numbers a, b, c between 2 and 18 such that: (i) their sum...

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  18. Three numbers, whose sum is 21, are in A.P. If 2, 2, 14 are added to t...

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  19. If X=1+a+a^(2)+a^(3)+"..."+infty " and " y=1+b+b^(2)+b^(3)+"..."+infty...

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  20. If S(1),S(2), S(3),......, S(p) are the sums of infinite geometric ser...

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