# limits of sequences

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• Oct 11th 2006, 01:04 PM
savetra
limits of sequences
a, Let q be the element of real numbers with q which is not equal to 0.
Show that if the absolute of q is less than 1, then
the limit of an absolute of q^n=0 as lim approaches positive infinity.

b, As the absolute of q which is greater than 1 then the
limit of an absolute of q^n =positive infinity as the limit approaches positive infinity.

c, What is the value of x if the sum of n=2 to positive infinity of (1+x)^-n=2
• Oct 11th 2006, 01:18 PM
ThePerfectHacker
Quote:

Originally Posted by savetra
a, Let q be the element of real numbers with q which is not equal to 0.
Show that if the absolute of q is less than 1, then
the limit of an absolute of q^n=0 as lim approaches positive infinity.

Use the ratio test,
lim |q|^{n+1}/|q|^n=|q|<1
Thus,
The infinite series,
SUM |q|^n---->Converges.
But then the sequence must converge to zero,
That it,
lim |q|^n---->0
----
b)

If |q|>1
Then multiplication by |q|>0 yields,
|q|^2>|q|
Again,
|q|^3>|q|^2
Again and again...
Thus the sequence is not bounded.