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sara_87
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Homework Statement
find:
lim(n[tex]\rightarrow\infty[/tex] (1/n^2 + 2/n^2 + 3/n^2 + ... + n-1/n^2 )
Homework Equations
3. The Attempt at a Solution [/b
I could guess that the limit is zero but i don't know howto prove it
sara_87 said:Homework Statement
find:
lim(n[tex]\rightarrow\infty[/tex] (1/n^2 + 2/n^2 + 3/n^2 + ... + n-1/n^2 )
Homework Equations
3. The Attempt at a Solution [/b
I could guess that the limit is zero but i don't know howto prove it
sara_87 said:I have done it and i just want to make sure of my answer:
i used the summation formula for aritmetic series :1/2 n(n+1)
and got that the limit is 1/2
is that right?
Dick said:BTW, I didn't say anything before in this thread. That was Halls. Let's give credit where credit is due.
Mark44 said:Sorry for giving credit where not due. I think I saw your name in another thread that sara started, and mistakenly cited you and Dirk.
The expression represents a mathematical series that converges to a finite value as n approaches infinity. This means that as n increases, the sum of the terms in the series approaches a limit value.
To calculate the limit, we can use the formula for the sum of a finite geometric series: S = a(1-r^n)/(1-r), where a is the first term, r is the common ratio, and n is the number of terms. In this case, a = 1, r = 1/n, and n = n-1. Plugging in these values and taking the limit as n approaches infinity, we get S = 1/(1-1/n) - 1 = 1 - 1 = 0.
The limit represents the end behavior of the series as n approaches infinity. It tells us the value that the series will approach and helps us understand the overall behavior of the series.
No, the limit of the series will always be a positive value. This is because as n increases, the terms in the series become smaller and smaller, approaching zero. Therefore, the sum of these terms will also approach zero, which is a positive value.
The limit of this series, 0, is considered a convergent limit, meaning that the series approaches a finite value as n-> infinity. This is in contrast to a divergent limit, where the series does not approach a finite value. In comparison to other infinite series, this series converges faster than many other commonly studied series, such as the harmonic series.