Solving Sequences & Series: Limits & Sums

[Bashyboy]

Hello,

I am curious to know that if we take some seqence, [itex]a_n[/itex], and take the limit as the the terms of the sequence goes to infinity, will the sequence head towards the same value that the the sum of the infinite amount of terms added together? (I hope I worded that properly...)

 

[Bashyboy]

I have another question, regarding infinite series. The book I am reading says if we want to find the sum of an infinite series, we have to consider the following sequence of partial sums. (I attached the picture.) What is the point of introducing this idea, does it elucidate anything? because it seems to just confuse me--and they don't even elaborate on why they are doing this. Could someone please help by explaining it to me?

 

[Mmm_Pasta]

Take the sequence 1/2, 1/4/ 1/8, 1/16, ... (1/2)^n. The sequence converges to 0, but does the sum converge to 0? We can clearly see that it does not.

To answer your second post. It seems you are studying geometric series at the moment which I gave you an example of. One can derive a formula for finding the sum of a geometric series by considering partial sums.

 

[SteveL27]

Hello,

I am curious to know that if we take some seqence, [itex]a_n[/itex], and take the limit as the the terms of the sequence goes to infinity, will the sequence head towards the same value that the the sum of the infinite amount of terms added together? (I hope I worded that properly...)

If a series converges at all, it's necessary for the corresponding sequence of terms to converge to zero. Mmm_Pasta gave an example.

Note that a sequence converging to zero does not necessarily mean that the series converges. The famous harmonic series 1/2 + 1/3 + 1/4 + 1/5 + ... diverges to infinity; yet the corresponding sequence of terms goes to zero.

I have another question, regarding infinite series. The book I am reading says if we want to find the sum of an infinite series, we have to consider the following sequence of partial sums. (I attached the picture.) What is the point of introducing this idea, does it elucidate anything? because it seems to just confuse me--and they don't even elaborate on why they are doing this. Could someone please help by explaining it to me?

The axioms of the real numbers tell us that we can add two numbers and get a third. And we can extend that to work with any finite sum of terms.

But how can we even assign meaning to an infinite sum? We do that by first defining the limit of a sequence. Then, given an infinite sum, we form the sequence of partial sums; and we say that the series converges if the sequence of partial sums does.

 

[blue_raver22]

Hello,

That is a great question! When dealing with sequences and series, we often encounter the concept of limits and sums. The limit of a sequence, as the terms approach infinity, represents the value towards which the sequence is heading. In other words, it is the value that the terms of the sequence get closer and closer to, but may never actually reach. On the other hand, the sum of an infinite series is the result of adding an infinite number of terms together. In some cases, the limit of a sequence may be equal to the sum of the infinite series. This is known as a convergent series, and it means that the terms of the sequence are getting closer and closer to a finite value. However, there are also cases where the limit and sum may not be equal, and this is known as a divergent series. In this case, the terms of the sequence may be getting closer to a value, but it is not a finite value and the sum of the infinite series does not exist. It is important to carefully consider the behavior of the terms of the sequence and the properties of the series in order to determine whether the limit and sum are equal or not. I hope this helps to clarify the relationship between limits and sums in sequences and series.