Recursive sequence terms don't cancel

In summary, the conversation discusses a telescopic series and finding the exact sum by performing a partial fraction decomposition and generalizing the formula for the nth partial sum Sn. The solution involves solving for the coefficients A, B, and C and the final formula for Sn includes terms with fractions and odd numbers. The mistake of forgetting a minus is also noted and corrected.
  • #1
whompa
2
0

Homework Statement



The following series is a telescopic series. Find the exact sum of the series by performing a partial fraction decomposition and generalizing the formula for the nth partial sum Sn.

problem.png


Homework Equations


The Attempt at a Solution



3n + 2 = A(n+1)(n+2) + B(n)(n+2) + C(n)(n+1)
n = 0 → 2 = 2A → A = 1
n = -1 → -1 = -B → B = 1
n = -2 → -4 = 2C → C = -2

(3n+2)/(n(n+1)(n+2)) = 1/n + 1/(n+1) + 2/(n+2)

=> Sn = ... = 1 + 1/2 + 1/3 + ... + 1/n + 2/3 + 2/5 + 2/7 + ... + 1/(2n + 1)

I don't see the terms cancel. Did i do something wrong?

Thanks
 
Last edited:
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  • #2
You forgot about the minus, dude! C=-2 :D
 
  • #3
Ah, sorry about that, i did have a minus in my original work, typo o_o
 

What is a recursive sequence?

A recursive sequence is a sequence of numbers where the next term is defined by a function or equation that involves previous terms in the sequence.

Why don't recursive sequence terms cancel?

Recursive sequences are designed to build upon each other, meaning that each term relies on the previous term in order to be calculated. This dependency on previous terms is what prevents the terms from canceling out.

Can a recursive sequence have negative terms?

Yes, a recursive sequence can have negative terms. The function or equation used to generate the sequence may include operations that result in negative numbers.

How are recursive sequence terms useful?

Recursive sequences can be used to model real-life situations, such as population growth or compound interest. They can also be used in mathematical proofs and to solve equations.

What is the difference between a recursive sequence and a series?

A recursive sequence is a list of numbers that follow a specific pattern, while a series is the sum of all the terms in a sequence. In other words, a series is a representation of a recursive sequence as a single number.

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