find the nth term awmd partail sum of the harmonic series below
1/3+1/8+1/15+1/12+..........show working plz
3 = 1*3
8 = 2*4
15 = 3*5
You sure 1/12 is next? Not
1/24, since
24 = 4*6
Partial sum =
1/2{3/2-2N+3/(N+1)(N+2)}
To find the nth term and the partial sum of the given harmonic series, we need to examine the pattern in the series and apply some mathematical concepts.
The given harmonic series is: 1/3 + 1/8 + 1/15 + 1/12 + ...
First, let's write down the fractions in the series in a simplified form.
1/3 can be written as 1/(3*1).
1/8 can be written as 1/(2^3 * 1).
1/15 can be written as 1/(3 * 5).
1/12 can be written as 1/(2^2 * 3).
From these examples, we can observe a pattern. The denominators are multiples of consecutive natural numbers, starting from 1. Let's analyze this pattern further.
The nth denominator in the series can be found using the formula: n * (n + 1)
To find the nth term of the series, we can express it as 1 divided by the nth denominator:
1/n * (n + 1)
Now, let's calculate the values of the nth terms for a few values of n:
When n = 1: 1/1 * (1 + 1) = 1/2
When n = 2: 1/2 * (2 + 1) = 3/2
When n = 3: 1/3 * (3 + 1) = 4/3
When n = 4: 1/4 * (4 + 1) = 5/4
So, the nth term of the series can be expressed as:
nth term = 1/n * (n + 1)
Next, let's find the partial sum of the series.
To calculate the partial sum, we need to add up all the terms in the series up to a certain term.
The sum of a series up to the nth term can be found using the formula: S_n = Sum of first n terms = (n/1) * ((n + 1)/2)
Let's calculate the partial sums for a few values of n:
When n = 1: Sm_1 = (1/1) * ((1 + 1)/2) = 1/2
When n = 2: Sm_2 = (2/1) * ((2 + 1)/2) = 3/1 = 3
When n = 3: Sm_3 = (3/1) * ((3 + 1)/2) = 6/1 = 6
When n = 4: Sm_4 = (4/1) * ((4 + 1)/2) = 10/1 = 10
So, the partial sum of the series can be expressed as:
Sm_n = (n/1) * ((n + 1)/2)
To find the nth term and the partial sum for a specific value of n, simply substitute that value into the respective formulas above.