Question

Determine whether each sequence converges or diverges If it converges, give the limit.
1) fn=(-3)^n
2) gn=(-1/3)^n

Answer 1

these are both 'geometric' sequences

the first sequence is
(-3)^1 , (-3)^2 , (-3)^3 , ...
=
-3, 9, -27, 81, ... does this sequence converge?

Answer 2

Since it continues to grow, yes? I'm not 100% sure on how to tell if a sequence converges or not.

Answer 3

the terms alternate
they do not get closer to a number

Answer 4

So, that means it diverges?

Answer 5

\[\text{ example } \\ \text{ Let } a_n=\frac{n-1}{n+1} \\ \lim_{n \rightarrow \infty} a_n=1 \\ \text{ so } a_n \text {converges to } 1 \]

Answer 6

a_n converges because the numbers in the sequence get closer to a number as our n gets larger

Answer 7

Okay, so number one definitely wouldn't converge because it isn't constant.

Answer 8

\[-1,0,\frac{1}{3},\frac{2}{4}, \frac{3}{5},\frac{4}{6},\frac{5}{7},\frac{6}{8},\frac{7}{9},\frac{8}{10},...\]
still a_n
a_n converges because it gets closer to a particular number in this that number is 1
the sequence isn't a constant

Answer 9

or this paritcular sequence isn't a constant sequence
but if we did have a sequence like
\[b_n=5 \\ \text{ then we say} b_n \text{ converges to } 5 \]

Answer 10

do you think the second one you have converges or diverges?

Answer 11

now the numbers do alternate
but do they alternate far from a certain number or close to a certain number

Answer 12

It converges because it continues to increase, correct?

Answer 13

Or since it is negative it would actually be decreasing.

Answer 14

that isn't what i'm saying...

Answer 15

that is a picture of the second one
but i think we need to go back to the first because i think you still don't understand
but first since @ganeshie8 was so kind to graph the second one

Answer 16

what do the dots look like they are close to as n gets large