foo9008 said: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 ) ?
Homework Equations
The Attempt at a Solution
in my opinion , it's also harmonic series , because the sum is divergent . Am i right ?
i mean second one , IMO , it is also harmonic ... , am i right ?Ray Vickson said: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.)
foo9008 said:i mean second one , IMO , it is also harmonic ... , am i right ?
A harmonic series is a type of mathematical series where the terms are reciprocals of positive integers. It is represented by the formula ∑(1/ k), where k is the positive integer.
A divergent harmonic series means that the sum of its terms approaches infinity as the number of terms increases. In other words, the series does not have a finite sum or limit.
To determine if a harmonic series is divergent, you can use the integral test or the comparison test. The integral test involves finding the value of the improper integral of the series, while the comparison test compares the series to a known divergent or convergent series.
Yes, the harmonic series ∑(1/ k+1 ) is divergent. This can be proven using the integral test, where the integral of the series is ln(x), which diverges as x approaches infinity.
No, a harmonic series can never be convergent. This is because as the terms of the series get smaller, the series still approaches infinity due to the infinite number of terms. In other words, the series does not have a finite limit.