Harmonic series

Homework Statement

i know that k = 0 to∞∑(1/ k) is harmonic series( we know that the sum is divergent) , how about ∑(1/ k+1 ) ???

The Attempt at a Solution

in my opinion , it's also harmonic series , because the sum is divergent . Am i right ?

Ray Vickson
Homework Helper
Dearly Missed

Homework Statement

i know that k = 0 to∞∑(1/ k) is harmonic series( we know that the sum is divergent) , how about ∑(1/ k+1 ) ???

The Attempt at a Solution

in my opinion , it's also harmonic series , because the sum is divergent . Am i right ?
Note: the series cannot be ##\sum_{k=0}^{\infty} 1/k## because the first term would be 1/0. However, starting at ##k = 1## is OK.

Do you mean that one of the series is ##\sum_{k=1}^{\infty} 1/k## and the other is ##\sum_{k=1}^{\infty}(\frac{1}{k}+1)##? That is what you wrote. Did you really mean ##\sum_{k=1}^{\infty} 1/(k+1)## for the second series? If so, use parentheses, like this: 1/(k+1).

Anyway, just write out the first first few terms of both of your series, to see how they are related.

When you say "it's also harmonic series , because the sum is divergent" you have it backwards: it is not harmonic because it is divergent; it is divergent because it is harmonic. (Lots of divergent series are not at all harmonic.)

Note: the series cannot be ##\sum_{k=0}^{\infty} 1/k## because the first term would be 1/0. However, starting at ##k = 1## is OK.

Do you mean that one of the series is ##\sum_{k=1}^{\infty} 1/k## and the other is ##\sum_{k=1}^{\infty}(\frac{1}{k}+1)##? That is what you wrote. Did you really mean ##\sum_{k=1}^{\infty} 1/(k+1)## for the second series? If so, use parentheses, like this: 1/(k+1).

Anyway, just write out the first first few terms of both of your series, to see how they are related.

When you say "it's also harmonic series , because the sum is divergent" you have it backwards: it is not harmonic because it is divergent; it is divergent because it is harmonic. (Lots of divergent series are not at all harmonic.)
i mean second one , IMO , it is also harmonic .... , am i right ?

Ray Vickson