# Thread: does this sum exist?

1. ## does this sum exist?

Could someone please help me to show whether or not the sum from k=1 to infinity of 1/2k exist or doesn't exist.

2. Originally Posted by ravenboy
Could someone please help me to show whether or not the sum from k=1 to infinity of 1/2k exist or doesn't exist.
$\displaystyle \sum_{k=1}^{+\infty} \frac{1}{k}$ is the famous harmonic series (Google it).

3. it doesn't.

see http://en.wikipedia.org/wiki/Harmoni...s_(mathematics).

(ahh, got beaten.)

4. basically i am trying to prove the sum of k=1 to infinity of 1/(K+ sqrt(k)) doesn't exist.
i know the sequence is greater than or equal to 1/2k and 1/2k doesn't exist so by comparison test our sequence doesn't exist.
how would you show 1/2k doesn't exist? Is it because:
we know 1/k doesn't exist and so 1/2k is just (1/2)*(1/k) so 1/2k doesn't exist.
Am i right in, saying we cant use comparison test to show 1/2k doesnt exist compared to 1/k because (1/2k)<(1/k). thanks for your help thus far.

5. Originally Posted by ravenboy
basically i am trying to prove the sum of k=1 to infinity of 1/(K+ sqrt(k)) doesn't exist.
i know the sequence is greater than or equal to 1/2k and 1/2k doesn't exist so by comparison test our sequence doesn't exist.
how would you show 1/2k doesn't exist? Is it because:
we know 1/k doesn't exist and so 1/2k is just (1/2)*(1/k) so 1/2k doesn't exist. Mr F says: Yes.

Am i right in, saying we cant use comparison test to show 1/2k doesnt exist compared to 1/k because (1/2k)<(1/k). thanks for your help thus far. Mr F says: Why would you be trying to do this when your previous argument is fine ....
....

6. thanks. wasn't sure if the following bit
""1/k doesn't exist and so 1/2k is just (1/2)*(1/k) so 1/2k doesn't exist."
was true or not as lecturer didn't really explain that bit, So i was assuming that was true. Thanks again.

7. Any nonzero constant multiple times a sum doesn't change whether it converges or not.