arithmetic and geometric series and sequences

[lvrgl31]

Would someone be kind enough to help me with this?
[i]Using the index of a series as the domain and the value of the series as the range, is a series a function? Include in the answer: a) Which one of the basic functions (linear, quadratic, rational, or exponential) is related to the arithmetic series. This one is linear, right? b) Which one of the basic functions is related to the geometric series? This one is exponential, right?c) Give real-life examples of both arithmetic and geometric sequences and series. Explain how these examples mightaffect you personally. An example I think is arithmetic is figuring the value of a person's assets.
I just need more help with the others. Any help would be appreciated. I'm still a little confused with series and sequences.

 

[pka]

Using the index of a series as the domain and the value of the series as the range, is a series a function?

The quick answer is no it is not a function.
The series \(\displaystyle \sum\limits_{k = J}^\infty {a_k }\) may or may not converge. If it converges it is a number and not a function.

But having said that, I think that whoever wrote the question may have had the idea of a sequence in mind. The classic definition of a sequence is: A sequence is a function from the positive integers to the real numbers.
Then given any series such as \(\displaystyle \sum\limits_{k = 1}^\infty {a_k }\) we define \(\displaystyle S_n = \sum\limits_{k = 1}^n {a_k }\) to be a sequence of partial sums associated with the series.

The we say that the \(\displaystyle \sum\limits_{k = 1}^\infty {a_k }\) converges if and only if its associated sequence of partial sums converges.

So you see there is some sense in which a convergent series is a function in that its associated sequence is a function.

 

[lvrgl31]

Thank you so much for your help.