harmonic series:1/3+1/8+1/15+1/12+.......
find the nth term and partail sum of the harmonic series:1/3+1/8+1/15+1/12+.......
I do not know what sort of harmonic series that is. None I know of.
You sure that last value is 1/12, and not 1/24?
oh yes yes 1/24
Careful proofreading would have saved you, Damon, and Steve a lot of time.
1/(3*1) + 1/(4*2) + 1/(5*3) + 1/(6*4)
1/[(n+2)n] or 1/(n^2+2n)
The given series is known as the harmonic series. To find the sum of this series, let's break down the terms and observe any patterns.
The general term of the series is given by 1/n(n + 2), where n takes the values 1, 2, 3, 4, ...
So, let's write down the terms of the series:
Term 1: 1/1(1 + 2) = 1/3
Term 2: 1/2(2 + 2) = 1/8
Term 3: 1/3(3 + 2) = 1/15
Term 4: 1/4(4 + 2) = 1/24
...
and so on.
Now, notice a pattern in the denominators:
The denominator is the product of two terms n and (n + 2), which can be simplified as n(n + 2). Also, notice that the numerator is always 1.
So, the simplified form of the general term is 1/n(n + 2).
To find the sum of the series, we need to add up all the terms. Let's express the terms in the simplified form:
1/3 + 1/8 + 1/15 + 1/24 + ...
Now, we'll use a concept known as partial fraction decomposition to simplify the series further.
By using partial fraction decomposition, we can express the general term as a sum of simpler fractions. In this case, we express 1/n(n + 2) as (1/2n) - (1/2(n + 2)).
So, the series can be written as:
[(1/2 * 1) - (1/2 * 3)] + [(1/2 * 2) - (1/2 * 4)] + [(1/2 * 3) - (1/2 * 5)] + ...
Now, notice that a pattern has emerged:
[(1/2 * 1) - (1/2 * 3)] can be written as (1/2) - (3/2)
[(1/2 * 2) - (1/2 * 4)] can be written as (1/2) - (1)
[(1/2 * 3) - (1/2 * 5)] can be written as (1/2) - (2/2)
We can observe that each term can be simplifed to (1/2) - (n/2), where n takes the values 1, 2, 3, 4, ...
Therefore, the sum of the series can be written as:
(1/2) - (1/2) + (1/2) - (2/2) + (1/2) - (3/2) + ...
Now, simplify further:
Combine the like terms: (1/2 - 1/2) + (1/2 - 2/2) + (1/2 - 3/2) + ...
This simplifies to:
0 + (-1/2) + (-2/2) + (-3/2) + ...
Now, add up the terms:
0 - (1/2) - (2/2) - (3/2) - (4/2) - (5/2) - ...
This can be simplified to:
- (1 + 2 + 3 + 4 + 5 + ...)/2.
The sum of the natural numbers 1, 2, 3, 4, 5, ... is given by the formula n(n + 1)/2, where n is the last term. In this case, the last term is infinity. Therefore, the sum of the natural numbers is infinity.
Substituting this into the expression, we get:
- (infinity)/2.
So, the sum of the given harmonic series is negative infinity.
Note: The harmonic series is a divergent series, meaning that its sum does not converge to a finite value. In other words, it keeps getting larger and larger as more terms are added.