Difference between convergence of partial sum and series

In summary, the notation Ʃan refers to the convergence of the sum of a series, or the convergence of its sequence of partial sums. As for its physical meaning, there is no standard interpretation, but if the series converges, the added terms must become infinitesimally small.
  • #1
Teachme
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I am a little confused as to notation for convergence. I included a picture too.
If you take a look it says "then the series Ʃan is divergent"
Does the "Ʃan" just mean the convergence as to the sum of the series, or the lim an as n→ ∞ nth term?
I believe it is the sum of the series but I just want to make sure.

As for the physical meaning of Ʃan I think that's related to the area, but what is the physical meaning of taking the lim an as n→ ∞ for the nth term?
Thanks
 

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  • #2
You are correct that when one talks about the convergence of a series Ʃan one is referring to the convergence of its sum, or put more eloquently, the convergence of its sequence of partial sums.

As for its physical interpretation, I'm not aware of any canonical physical interpretation of infinite series in general. However, if the series converges, that is to say if its sequence of partial sums converges, then the bits added to the sum must become infinitesimally small, although this is not a sufficient requirement for its convergence; the standard counter-example being Ʃ1/n.
 
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1. What is the difference between convergence of partial sum and series?

The convergence of partial sum refers to the sum of a finite number of terms in a series, while the convergence of a series refers to the behavior of the entire infinite series. In other words, the convergence of a partial sum is a local property, while the convergence of a series is a global property.

2. How is convergence of partial sum and series determined?

The convergence of a partial sum can be determined using the partial sum test, which states that if the sequence of partial sums converges, then the series also converges. The convergence of a series can be determined using various tests such as the ratio test, comparison test, or integral test.

3. Can a series converge if its partial sums do not?

Yes, it is possible for a series to converge even if its partial sums do not. This is because the behavior of the entire series is determined by the infinite number of terms, rather than just a finite number of terms in the partial sum.

4. What is the significance of convergence of partial sum and series?

The convergence of a series is important in determining the sum of the infinite series. It also helps in understanding the behavior of a series and its relationship to other mathematical concepts such as integrals and derivatives. The convergence of partial sum is useful in approximating the sum of a series by considering a finite number of terms.

5. How does the rate of convergence differ between partial sum and series?

The rate of convergence for a series is typically faster than that of its partial sums. This is because the partial sums only consider a finite number of terms, while the series takes into account the behavior of all terms in the infinite series. Therefore, the convergence of a series is a stronger indication of the behavior of the series as a whole.

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