Question

Find the sum of the following harmonic series: $\dfrac{1}{2} + \dfrac{1}{4} + \dfrac{1}{6} + ..........10$ terms.

Answer 1

Hint- If $\dfrac{1}{a},\dfrac{1}{{a + d}},\dfrac{1}{{a + 2d}}............$ is given harmonic progression, the formula to find the sum of n terms in the harmonic progression is given by the formula
Complete step-by-step answer:
$
  {\text{sum of n terms}} \\
  {{\text{S}}_n} = \dfrac{1}{d}\ln \left\{ {\dfrac{{2a + \left( {2n - 1} \right)d}}{{2a - d}}} \right\} \\
 $
Where
a is the first term of H.P
d is the common difference of H.P
ln is the natural logarithm
The given series is
$\dfrac{1}{2} + \dfrac{1}{4} + \dfrac{1}{6} + ..........$
We have to find the sum of first 10 terms
So n = 10, a = 2
The value of d is
$4 - 2 = 2$
The sum of H.P is given by
$
  {\text{sum of n terms}} \\
  {{\text{S}}_n} = \dfrac{1}{d}\ln \left\{ {\dfrac{{2a + \left( {2n - 1} \right)d}}{{2a - d}}} \right\} \\
 $
Substituting the values of a, n and d from above
$
  {\text{sum of 10 terms}} \\
  {{\text{S}}_{10}} = \dfrac{1}{d}\ln \left\{ {\dfrac{{2a + \left( {2n - 1} \right)d}}{{2a - d}}} \right\} \\
   \Rightarrow {{\text{S}}_{10}} = \dfrac{1}{2}\ln \left\{ {\dfrac{{2\left( 2 \right) + \left( {2\left( {10} \right) - 1} \right)2}}{{2\left( 2 \right) - \left( 2 \right)}}} \right\} \\
   \Rightarrow {{\text{S}}_{10}} = \dfrac{1}{2}\ln \left\{ {\dfrac{{4 + 19 \times 2}}{{4 - 2}}} \right\} \\
   \Rightarrow {{\text{S}}_{10}} = \dfrac{1}{2}\ln \left\{ {\dfrac{{4 + 38}}{{4 - 2}}} \right\} \\
   \Rightarrow {{\text{S}}_{10}} = \dfrac{1}{2}\ln \left\{ {\dfrac{{42}}{2}} \right\} \\
   \Rightarrow {{\text{S}}_{10}} = \dfrac{1}{2}\ln \left( {21} \right) \\
 $
Now, let us substitute the value of natural logarithm of 21 from the table.
As we know from the table $\ln \left( {21} \right) = 3.0445$
So the sum of the series is
$
   \Rightarrow {{\text{S}}_{10}} = \dfrac{1}{2}\ln \left( {21} \right) = \dfrac{1}{2} \times 3.0445 \\
   \Rightarrow {{\text{S}}_{10}} = 1.5222 \\
 $
Hence, the sum of 10 numbers of given harmonic series is 1.5222

Note- A harmonic progression is a progression formed by taking the reciprocals of an arithmetic progression. Equivalently, a sequence is a harmonic progression when each term is the harmonic mean of the neighboring terms. Students must remember the formula for the sum of n terms of a harmonic progression.