Proof by Induction - Requires calculus

In summary, the conversation discusses a question that requires a proof using calculus. The question involves expanding (1+x)n using the binomial theorem and using differentiation and integration. The conversation ends with someone suggesting to take the first derivative and substituting x=1 to get the desired result.
  • #1
mathsgeek
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[SOLVED] Proof by Induction - Requires calculus

I have received a question that i have been trying to do and requires a proof to prove that it is true. The question is here:

http://img401.imageshack.us/my.php?image=23688074ji0.png

I am having some trouble to prove this as at some stage u are required to use differentiation and integration however i do not know homework to accomplish this, could somebody help me get on the right track and start me off for this question as i am having moderate difficulty? thankyou
 
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  • #2
Start with expanding (1+x)n using the binomial theorem and get
1) (1+x)n=c0+c1x+...cnxn
Take the first derivative and get
2) n(1+x)n-1=c1+...ncnxn-1

Finally, let x=1 and add 1 and 2 together to get the desired result.
 
  • #3
Please delete thread, thankyou
 
Last edited:

1. What is proof by induction and how does it relate to calculus?

Proof by induction is a mathematical technique used to prove that a statement holds for all natural numbers. It involves proving a base case and an inductive step, which together show that the statement holds for all natural numbers. This technique is often used in calculus to prove properties of functions and sequences.

2. Why is calculus needed for proof by induction?

Calculus is needed for proof by induction because it involves dealing with functions and sequences, which are fundamental concepts in calculus. Calculus provides the tools and methods to analyze and manipulate these mathematical objects, making it an essential part of the proof by induction process.

3. What is the difference between mathematical induction and proof by induction?

Mathematical induction is a general proof technique used to prove statements about all natural numbers. Proof by induction is a specific application of mathematical induction, where the statement being proved is related to calculus and requires the use of calculus techniques.

4. Can you give an example of proof by induction that requires calculus?

One example of proof by induction that requires calculus is the proof of the sum of the first n natural numbers, given by the formula n(n+1)/2. This proof involves using calculus techniques, such as integration and differentiation, to manipulate the formula and show that it holds for all natural numbers.

5. Are there any limitations to proof by induction with calculus?

One limitation of proof by induction with calculus is that it only applies to statements that can be expressed using calculus concepts. This means that it may not be applicable to all mathematical statements, and other proof techniques may need to be used. Additionally, proof by induction may not work for infinite sequences or series, as calculus techniques may not be able to handle them.

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