# Give examples of sequences;Determine which of the following series converges and comp

• Jun 2nd 2010, 12:12 AM
ewkimchi
Give examples of sequences;Determine which of the following series converges and comp
A)
Give examples of sequences:
a) one that converges to 100
b) one that converges to 3/4
c) one that diverges

Justify what you are presenting.

B)
Determine which of the following series converges and compute its sum:
a) n=0, infinite 1/n^(1/3)
b) n=0, infinite 4^n

• Jun 2nd 2010, 04:56 AM
mr fantastic
Quote:

Originally Posted by ewkimchi
A)
Give examples of sequences:
a) one that converges to 100
b) one that converges to 3/4
c) one that diverges

Justify what you are presenting.

B)
Determine which of the following series converges and compute its sum:
a) n=0, infinite 1/n^(1/3)
b) n=0, infinite 4^n

What have you tried here? Where are you stuck?

For A), note that 1/n --> 0 ....

For B), neither series converges. Your job now is to think about why.
• Jun 2nd 2010, 09:52 AM
ewkimchi
Did I get it right?
Do these work?

A)
a) (99n + 1)/100n

b) (3n+1)/4n

c) (n^3)/(n^2 + 1)

B)
a)
by p series test, diverges b/c 1/3<1

b)
by geometric series test, converges b/c 1/4<1 and sum=8/3

c)
I'm not sure about this one because 4 isn't under a fraction or does that even matter?
Anyway, I got
by geometric series test diverges b/c 4>1
• Jun 2nd 2010, 10:42 AM
General
Quote:

Originally Posted by ewkimchi
Do these work?

A)
a) (99n + 1)/100n
No, this one converges to $\color{blue} \frac{99}{100}$.
b) (3n+1)/4n
Correct.
c) (n^3)/(n^2 + 1)
Correct.

B)
a)
by p series test, diverges b/c 1/3<1
Correct.
b)
by geometric series test, converges b/c 1/4<1 and sum=8/3
The series in your question is 4^n not (1/4)^n !
c)
I'm not sure about this one because 4 isn't under a fraction or does that even matter?
Anyway, I got
by geometric series test diverges b/c 4>1
There is no part c in your question !
And remember that, you should check the absolute value of the common ratio.
For instance,
$\sum_{n=0}^{\infty} \left(\frac{-3}{2}\right)^n$ DIVERGES.
because |-3/2|=3/2>1.

Anyway:

For $\color{blue} \sum_{n=0}^{\infty} 4^n$ : The series diverges by the nth term test(Test for Divergence).

For $\color{blue} \sum_{n=0}^{\infty} \left(\frac{1}{4}\right)^n$ : The series converges b/c its a geomreic series with common ration |1/4|=1/4< 1.