Question

calculus- how can i tell if a series is an alternating series by looking at it?

Answer 1

\[(-1)^n\] is usually there

Answer 2

will there always be a negative number?

Answer 3

just minus one to the n or n plus one

Answer 4

but it could be something like \(\sum \frac{(-2)^n}{3^n}\) say

Answer 5

on the other hand, you can always factor out the \((-1)^n\) so it is usually written that way

Answer 6

what if theres no exponent n in the numerator?

Answer 7

then no probably not
you have an example?

Answer 8

alternating of course means it goes \[a_1-a_2+a_3-a_4+...\] easiest way to write it is \((-1)^na_n\)

Answer 9

i'll show you an example in 2 min when the problem comes back (online hw)

Answer 10

kk

Answer 11

cos(pi*k) / k^2
(-1)^(2k) / k^2
(-1)^k*cos(k)

Answer 12

ok i see

Answer 13

some kind of trick

Answer 14

sin(k)
k / (-2)^k
(-1)^k / ln(k)

Answer 15

first off \((-1)^{2k}=1\) since \(2k\) is even and minus one to an even power is 1
so that one does not alternate

Answer 16

ok

Answer 17

i have never seen a question like this
i guess the question is "which ones alternate" right?

Answer 18

yes, mark each alternating series

Answer 19

cos(pi*k) / k^2
since \(\cos(\pi)=-1,\cos(2\pi)=1,\cos(3\pi)=-1\) etc, this is a tricky way to write an alternating series

Answer 20

(-1)^k*cos(k)
this one has \((-1)^k\) so it alternates for sure

Answer 21

sin(k) does not alternate it is just sine

Answer 22

and the last two do alternate, since they have either \((-1)^k\) or \((-2)^k\)

Answer 23

thank you.

Answer 24

so if there is (-#)^k its always alternating ?

Answer 25

very odd question
of course you could just check the first few terms to see
yea \((-\xi)^k\) alternates

Answer 26

pretty clear why right? raise a negative number to an even power, it is positive, raise to an odd number, it is negative

Answer 27

ok thank you

Answer 28

yw

Answer 29

i would be careful with \((-1)^k\cos(k)\)

since the cosine can be positive and negative also