Convergence Divergence Help

In summary, the conversation is about a student struggling with determining the convergence of improper integrals for their upcoming final exam. The student is specifically having trouble with the integrals -inf to 1 of: dx/(x-2)^5 and 0 to inf of: e^x dx. They have attempted to solve these integrals by splitting them into two parts and taking the limit, but have received conflicting answers and are seeking clarification. The responder advises against splitting the integrals and explains the correct approach to solving them.
  • #1
cooltee13
25
0

Homework Statement


Well I am studying for my final which is in a couple of days, and I am stuck on this topic of convergence of improper integrals. I've been doing a couple and I thought I was doing them correctly but my answers do not go with the answers I was given. So I am stressing out over it. Here are the questions I have:

Determine whether the following integrals converge or diverge, evaluate those that converge.

1) Integral from -inf to 1 of: dx/(x-2)^5

2) Integral from 0 to inf of: e^x dx


Homework Equations


Just the standard limit and integral equations


The Attempt at a Solution


1) I split the integral into two pieces, a) the int from -inf to 0: 1/(x-2)^5 and b) the int from 0 to 1: 1/(x-2)^5 I then took the limit of each integral, so I got:
a) lim as t approaches -inf of a. solving the limit I got (1/-2 - 1/inf) which = -1/2 and is convergant.

b) lim as t approaches 0 of b. solving that limit I came up with a divergent answer. making the original Integral divergent. however the answer I was given says that the integral is convergent. I am confused on this.
-------------------------------------------------------------------------
2) integral from 0 to inf of e^x dx
I split this integral into two pieces as well. a) integral from 0 to 1: e^x dx and b) integral from 1 to inf: e^x dx

Solving each part, I got two convergent answers, but the answer said that its divergent. I am wrong again :yuck:.

Please help me out, and explain why. I have the answers, that's not what I am going for..I just want to understand this so I can do well on my final. Thanks guys
 
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  • #2
I didn't get why you are splitting things.
You only do that when you have a discontinuity like going from -1 to 1 for 1/x.

1/(x-2)^2 .. gives -1/(x-2) ] from t to 1, and take limit of t as it appraches -inf

same thing for e^x
 
  • #3
b) the int from 0 to 1: 1/(x-2)^5 I then took the limit of each integral, so I got: ...

also, you did this step wrong!
you don't have any inf/discountinity from 0 to 1
 

1) What is convergence divergence?

Convergence divergence, also known as convergence divergence help or CDH, is a mathematical concept used in fields such as mathematics, physics, and engineering. It refers to the behavior of a series or sequence as it approaches a certain point or value. It is used to determine whether a series or sequence will continue to increase or decrease, or if it will eventually reach a steady state.

2) How is convergence divergence used in science?

Convergence divergence is used in various scientific fields to analyze and predict the behavior of systems and processes. For example, it can be used in physics to study the motion of objects, in chemistry to understand chemical reactions, and in biology to analyze the growth of populations. It is also commonly used in data analysis and forecasting in fields such as economics and finance.

3) What is the difference between convergence and divergence?

Convergence and divergence are two opposite outcomes of convergence divergence. Convergence refers to the behavior of a series or sequence as it approaches a certain point or value, while divergence refers to the behavior of a series or sequence as it moves away from a certain point or value. In convergence, the series or sequence approaches a steady state, while in divergence, it continues to increase or decrease without reaching a steady state.

4) How is convergence divergence calculated?

There are various methods for calculating convergence divergence, depending on the specific application and type of series or sequence being analyzed. Some common methods include using the limit of a sequence or series, using convergence tests such as the ratio test or root test, and using graphical methods such as plotting the terms of the series or sequence. Consult a textbook or mathematical resource for specific instructions on how to calculate convergence divergence for a particular problem.

5) Why is convergence divergence important in scientific research?

Convergence divergence is important in scientific research because it allows scientists to analyze and predict the behavior of systems and processes. By understanding how a series or sequence will behave, scientists can make informed decisions and draw conclusions about the underlying mechanisms and patterns. This can lead to advancements in various fields and contribute to the overall understanding of the natural world.

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