# What is the value of f'(1) for the series 1983 BC 5 part C?

• keithk
In summary, the value of f'(1) for the function f(x) = \sum^{\infty}_{n=0}a_{n}x^n, where a_{0} = 1 and a_{n} = (7/n)a_{n-1}, is 7e^7.
keithk
Sum of a Series - 1983 BC 5 part C

I don't have the actual problem (this is for a friend) but this is what I could gather from what she was saying.

## Homework Statement

If $$f(x) = \sum^{\infty}_{n=0}a_{n}x^n$$ find the value of f'(1)

$$a_{0} = 1$$ and $$a_{n} = (7/n)a_{n-1}$$

None maybe?

## The Attempt at a Solution

Ok so after differentiating,
$$f(x) = \sum^{\infty}_{n=1}n a_{n}x^{n-1}$$
Writing out the terms and subbing 1 for x got me to,
$$f(x) = \sum^{\infty}_{n=1}7^n/(n-1)!$$
or
$$f(x) = \sum^{\infty}_{n=1}n 7^n/(n)!$$

This was as far as I was able to get. Mathematica tells me that the answer is 7e^7.
I know that $$\sum^{\infty}_{n=1}n/(n)! = e$$ and that $$\sum^{\infty}_{n=0}7^n/(n)! = e^7$$ (which doesn't really help because we're
starting at 1). But with both parts in there I'm not sure what to do

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Ok I think I got it.
If $$f(x) = \sum^{\infty}_{n=0}a_{n}x^n$$ find the value of f'(1)
$$a_{0} = 1$$ and $$a_{n} = (7/n)a_{n-1}$$

$$f(x) = 1/0! + 7x/1! + 7^2x^2/2! + 7^3x^3/3!... \sum^{\infty}_{n=0}\frac{7^nx^n}{n!}$$
$$f'(x) = 0 + 7/0! + 7^2x/1! + 7^3x^2/2! ... \sum^{\infty}_{n=1}\frac{7^n x^{n-1}}{(n-1)!} = \sum^{\infty}_{n=1}\frac{n7^nx^{n-1}}{n!}$$
$$f'(1) = 7/0! + 7^2/1! + 7^3/2! ... \sum^{\infty}_{n=1}\frac{n7^n }{n!} = 7(1 + 7 + 7^2/2! + 7^3/3!...)$$
$$e^x = 1 + x + x^2/2! + x^3/3! +x^4/4!... \sum^{\infty}_{n=0}\frac{x^n }{n!}$$
$$e^7 = 1 + 7 + 7^2/2! + 7^3/3! + 7^4/4!... \sum^{\infty}_{n=0}\frac{7^n }{n!}$$
Recall that:
$$f'(1) = 7/0! + 7^2/1! + 7^3/2! ... \sum^{\infty}_{n=1}\frac{n7^n }{n!} = 7(1 + 7 + 7^2/2! + 7^3/3!...)$$
and compare to e^7:
Which gives:
7e^7

Sorry if that is a bit hard to follow.

Last edited:
Ohh I can't edit it anymore...I think it is correct but I wanted to change the forms of some stuff.

I like this better, for example:
$$f'(1) = 7/0! + 7^2/1! + 7^3/2! ... \sum^{\infty}_{n=1}\frac{7^n }{(n-1)!} = 7(1 + 7 + 7^2/2! + 7^3/3!...)$$

I got the same answer as you, 7e^7.

By the power series: $$e^7 = 1 + 7 + \frac{7^2}{2!} + \frac{7^3}{3!} + \frac{7^4}{4!} + \ ... \ \frac{7^n}{n!}$$.

This matches the expression for $$\frac{f'(1)}{7} = 1 + 7 + \ ... \ \frac{7^{n-1}}{(n-1)!}$$, except that it ends with n-1 rather n. However since both are infinite series, they are one and the same.

## 1. What is the significance of "1983 BC 5 part C" in history?

"1983 BC 5 part C" refers to a specific year and location in human history, which is the year 1983 BCE in the ancient city of Babylon. This time period is significant because it marks the beginning of the Babylonian Empire, one of the most powerful and influential empires in Mesopotamia.

## 2. What major events occurred during "1983 BC 5 part C"?

During "1983 BC 5 part C", the Babylonian Empire was established under the rule of Hammurabi. This was also the time when the Code of Hammurabi, a set of laws that governed the Babylonian society, was created. Additionally, many significant architectural and engineering projects were undertaken during this time, such as the construction of the Hanging Gardens of Babylon.

## 3. How did "1983 BC 5 part C" impact the development of human civilization?

The establishment of the Babylonian Empire in "1983 BC 5 part C" had a profound impact on the development of human civilization. The Code of Hammurabi set a precedent for written laws and government control, which influenced future legal systems. The Babylonian Empire also made significant advancements in mathematics, astronomy, and literature, which further contributed to the growth of human civilization.

## 4. Who were the major figures during "1983 BC 5 part C"?

The most notable figure during "1983 BC 5 part C" was Hammurabi, the sixth king of the Babylonian Empire. He is known for his military conquests, as well as the creation of the Code of Hammurabi. Other important figures during this time include Sargon of Akkad, who founded the Akkadian Empire, and Ur-Nammu, who established the Sumerian Third Dynasty of Ur.

## 5. What evidence do we have of "1983 BC 5 part C"?

There is physical evidence of the existence of the Babylonian Empire, such as the ruins of the city of Babylon and artifacts like the Code of Hammurabi. Additionally, there are historical records from other ancient civilizations, such as Egypt and Assyria, that mention the events and figures of "1983 BC 5 part C". Archaeological excavations and studies also provide further evidence of this time period and its impact on human civilization.

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