Advertisements
Advertisements
प्रश्न
If x is positive, the sum to infinity of the series \[\frac{1}{1 + x} - \frac{1 - x}{(1 + x )^2} + \frac{(1 - x )^2}{(1 + x )^3} - \frac{(1 - x )^3}{(1 + x )^4} + . . . . . . is\]
पर्याय
(a) 1/2
(b) 3/4
(c) 1
(d) none of these
Advertisements
उत्तर
(a) \[\frac{1}{2}\]
\[\text{ Let } S = \frac{1}{\left( 1 + x \right)} - \frac{\left( 1 - x \right)}{\left( 1 + x \right)^2} + \frac{\left( 1 - x \right)^2}{\left( 1 + x \right)^3} - \frac{\left( 1 - x \right)^3}{\left( 1 + x \right)^4} + . . . \infty \]
\[\text{ It is clear that it is a G . P . with a } = \frac{1}{\left( 1 + x \right)} \text{ and }r = - \frac{\left( 1 - x \right)}{\left( 1 + x \right)} . \]
\[ \therefore S = \frac{a}{\left( 1 - r \right)}\]
\[ \Rightarrow S = \frac{\frac{1}{\left( 1 + x \right)}}{\left[ 1 - \left( - \frac{\left( 1 - x \right)}{\left( 1 + x \right)} \right) \right]}\]
\[ \Rightarrow S = \frac{\frac{1}{\left( 1 + x \right)}}{\left[ 1 + \frac{\left( 1 - x \right)}{\left( 1 + x \right)} \right]}\]
\[ \Rightarrow S = \frac{\frac{1}{\left( 1 + x \right)}}{\left[ \frac{\left( 1 + x \right) + \left( 1 - x \right)}{\left( 1 + x \right)} \right]}\]
\[ \Rightarrow S = \frac{1}{2}\]
\[\]
APPEARS IN
संबंधित प्रश्न
Find the sum to indicated number of terms in the geometric progressions x3, x5, x7, ... n terms (if x ≠ ± 1).
Find the sum to n terms of the sequence, 8, 88, 888, 8888… .
if `(a+ bx)/(a - bx) = (b +cx)/(b - cx) = (c + dx)/(c- dx) (x != 0)` then show that a, b, c and d are in G.P.
Find the 4th term from the end of the G.P.
Which term of the G.P.: `sqrt3, 3, 3sqrt3`, ... is 729?
The fourth term of a G.P. is 27 and the 7th term is 729, find the G.P.
The sum of three numbers in G.P. is 14. If the first two terms are each increased by 1 and the third term decreased by 1, the resulting numbers are in A.P. Find the numbers.
Find the sum of the following geometric progression:
2, 6, 18, ... to 7 terms;
Find the sum of the following geometric progression:
(a2 − b2), (a − b), \[\left( \frac{a - b}{a + b} \right)\] to n terms;
Find the sum of the following geometric series:
`sqrt7, sqrt21, 3sqrt7,...` to n terms
Evaluate the following:
\[\sum^n_{k = 1} ( 2^k + 3^{k - 1} )\]
Find the sum of the following serie:
5 + 55 + 555 + ... to n terms;
How many terms of the series 2 + 6 + 18 + ... must be taken to make the sum equal to 728?
A person has 2 parents, 4 grandparents, 8 great grandparents, and so on. Find the number of his ancestors during the ten generations preceding his own.
Find the sum of the following serie to infinity:
8 + \[4\sqrt{2}\] + 4 + ... ∞
If a, b, c are in G.P., prove that the following is also in G.P.:
a2, b2, c2
If a, b, c, d are in G.P., prove that:
\[\frac{1}{a^2 + b^2}, \frac{1}{b^2 - c^2}, \frac{1}{c^2 + d^2} \text { are in G . P } .\]
If a, b, c, d are in G.P., prove that:
(a2 + b2 + c2), (ab + bc + cd), (b2 + c2 + d2) are in G.P.
If a, b, c are in A.P., b,c,d are in G.P. and \[\frac{1}{c}, \frac{1}{d}, \frac{1}{e}\] are in A.P., prove that a, c,e are in G.P.
If the fifth term of a G.P. is 2, then write the product of its 9 terms.
If pth, qth and rth terms of a G.P. re x, y, z respectively, then write the value of xq − r yr − pzp − q.
Mark the correct alternative in the following question:
Let S be the sum, P be the product and R be the sum of the reciprocals of 3 terms of a G.P. Then p2R3 : S3 is equal to
For the G.P. if r = `1/3`, a = 9 find t7
Find five numbers in G.P. such that their product is 1024 and fifth term is square of the third term.
Mosquitoes are growing at a rate of 10% a year. If there were 200 mosquitoes in the beginning. Write down the number of mosquitoes after 10 years.
The numbers x − 6, 2x and x2 are in G.P. Find nth term
For a G.P. sum of first 3 terms is 125 and sum of next 3 terms is 27, find the value of r
If the common ratio of a G.P. is `2/3` and sum to infinity is 12. Find the first term
Find : `sum_("r" = 1)^oo (-1/3)^"r"`
Find : `sum_("n" = 1)^oo 0.4^"n"`
Insert two numbers between 1 and −27 so that the resulting sequence is a G.P.
Select the correct answer from the given alternative.
The tenth term of the geometric sequence `1/4, (-1)/2, 1, -2,` ... is –
Answer the following:
For a G.P. a = `4/3` and t7 = `243/1024`, find the value of r
Answer the following:
For a sequence Sn = 4(7n – 1) verify that the sequence is a G.P.
Answer the following:
If for a G.P. t3 = `1/3`, t6 = `1/81` find r
If the pth and qth terms of a G.P. are q and p respectively, show that its (p + q)th term is `(q^p/p^q)^(1/(p - q))`
The sum of the first three terms of a G.P. is S and their product is 27. Then all such S lie in ______.
If 0 < x, y, a, b < 1, then the sum of the infinite terms of the series `sqrt(x)(sqrt(a) + sqrt(x)) + sqrt(x)(sqrt(ab) + sqrt(xy)) + sqrt(x)(bsqrt(a) + ysqrt(x)) + ...` is ______.
