Question

Show the series converges.

Answer 1

\[\sum_{1}^{\infty} x _{k}y _{k}\]
where
\[x _{k} = 1,1,-2,1,1,-2... and y _{k}= \frac{ 1 }{ \sqrt{k} }\]

Answer 2

k=1 for the summations

Answer 3

alternating series test?

Answer 4

I will look into that, trying to figure out if x_k can be written in a simpler form

Answer 5

try inserting bracket in 3 terms

Answer 6

@m_sequence1 can you write your problems a little larger? It's a little hard to see them :|

Answer 7

Sorry about that, I not sure how to make them bigger from the Equation editor here

Answer 8

/large before typing them up

Answer 9

\[/\large ^{}\]

Answer 10

/large \[x\]

Answer 11

I've been trying to make out what that letter was after x xD

Answer 12

I'm trying it right now lol\[/\large x\]

Answer 13

\[x ^{}\]

Answer 14

it worked

Answer 15

it was K

Answer 16

\[\large \sum_{k=1}^{\infty}X_{k}Y_{k}\]

Answer 17

Ohhh ok, sorry about that. I was just confused.

Answer 18

No need to worry, you showed me something new.

Answer 19

I new here and sometimes i feel people can type equations pretty fast

Answer 20

you must be careful when writing variables in caps, it often means matrix or set,
Ax means matrix * vector to me or cosets of A.

Answer 21

use
http://www.codecogs.com/latex/eqneditor.php

Answer 22

You can prove that \(\displaystyle\sum_{k=1} \frac{1}{\sqrt {3k}}+\frac{1}{\sqrt{3k+1}}\) is convergent.
Then you prove that:
\[\sum_{k=1}\frac{-2}{\sqrt{3k}}\]
Is also convergent. The sum of both series is the orginal series.

Answer 23

does the alternating series test not work?

Answer 24

@DanielM_113 both are not convergent series @zzr0ck3r alternating series test works ...just write it up as sum of two terms
\[ \sum_{n=1}^\infty \left ( \frac{1}{\sqrt 3n} + \frac{1}{\sqrt{3n+1}} - \frac{2}{\sqrt{3n+2}} \right ) = \sum_{n=1}^\infty \left ( \left( \frac{1}{\sqrt 3n} + \frac{1}{\sqrt{3n+1}} \right ) - \frac{2}{\sqrt{3n+2}} \right ) \]

Answer 25

also I think Cauchy test works, but a bit complicated one.

Answer 26

@experimentX Very good.