Revisiting the Convergence of Infinite Series in Calculus 2

In summary, the conversation discusses the relationship between the exponential and logarithmic functions and their corresponding series representations. It is stated that e^x is equal to the infinite sum of x^k/k!, and that ln(e^x) is equal to ln of the same series. It is then mentioned that x is equal to ln of the series. The validity and usefulness of this relationship is questioned, and it is noted that there may be a misunderstanding between the words "sense" and "since".
  • #1
GreenPrint
1,196
0
Sense e^x=Ʃ[k=0,∞] x^k/k!
then
ln(e^x) = ln(Ʃ[k=0,∞] x^k/k!)
x = ln(Ʃ[k=0,∞] x^k/k!)

is this true?
 
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  • #2
GreenPrint said:
Sense e^x=Ʃ[k=0,∞] x^k/k!
then
ln(e^x) = ln(Ʃ[k=0,∞] x^k/k!)
x = ln(Ʃ[k=0,∞] x^k/k!)

is this true?
Sure, but how useful it is, I don't know.

I sense that you don't understand the difference between sense and since.
 
  • #3
Mark44 said:
Sure, but how useful it is, I don't know.

I sense that you don't understand the difference between sense and since.

I'm not sure that it is useful at all and is why I asked lol. Nope my English is god awful.
 

1. What are infinite series in calculus 2?

An infinite series in calculus 2 is a sum of an infinite number of terms. It is represented by the notation Σ (sigma) and is used to approximate the value of a function or to find the sum of a sequence.

2. How do you determine the convergence of an infinite series?

The convergence of an infinite series can be determined by using various convergence tests such as the comparison test, ratio test, root test, and integral test. These tests involve comparing the given series to known convergent or divergent series and using their properties to determine the convergence of the given series.

3. What is the difference between a convergent and a divergent infinite series?

A convergent infinite series is one that has a finite sum or limit, meaning the terms in the series eventually get smaller and smaller and approach a specific value. On the other hand, a divergent infinite series is one that does not have a finite sum or limit, meaning the terms in the series do not approach a specific value and the series either grows infinitely or oscillates.

4. How is the sum of an infinite series calculated?

The sum of an infinite series is calculated by taking the limit of the partial sums of the series as the number of terms approaches infinity. This process is known as finding the limit of the sequence of partial sums, and the resulting value is the sum of the infinite series.

5. How is calculus 2 used in real-world applications?

Calculus 2, specifically infinite series, is used in various real-world applications such as physics, engineering, and economics. It is used to model and analyze continuous systems, such as motion, population growth, and economic trends, where the variables are constantly changing. It also plays a crucial role in signal processing, control theory, and optimization problems.

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