Question

The sum of the series $ \left( 1+\frac{{{({{\log }_{e}}\,n)}^{2}}}{2!}+\frac{{{({{\log }_{e}}n)}^{4}}}{4!}+... \right) $ is

• A. $ n+\frac{1}{n} $
• B. $ \frac{n'}{2}+\frac{1}{2n} $
• C. $ {{\log }_{e}}\frac{1}{1-{{({{\log }_{e}}n)}^{2}}} $
• D. $ \frac{1}{2}{{\log }_{e}}\,\frac{1}{1-{{({{\log }_{e}}\,n)}^{2}}} $

Answer 1

The Correct Option is B

We know that, $ 1+\frac{{{x}^{2}}}{2!}+\frac{{{x}^{4}}}{4!2}+....=\frac{{{e}^{x}}+{{e}^{-x}}}{{}} $
$ \therefore $ $ a+\frac{{{({{\log }_{e}}n)}^{2}}}{2!}+\frac{{{({{\log }_{e}}n)}^{4}}}{4!}+.... $
$ =\frac{{{e}^{{{\log }_{e}}n}}+{{e}^{-{{\log }_{e}}n}}}{2} $
$ =\frac{n+\frac{1}{n}}{2} $
$ =\frac{n}{2}+\frac{1}{2n} $

Answer 2

In mathematics, a series describes the process of successively adding an unlimited number of quantities to a specified initial quantity. Calculus and its generalization, and mathematical analysis, both include studying series as an essential component. Most branches of mathematics employ series, including combinatorics, where generating functions are used to explore finite structures. In addition to being widely utilized in mathematics, infinite series are also used extensively in physics, computer science, statistics, and finance, among other quantitative fields. On the other hand, the definition of a sequence is an enumerated collection of things where repeats are permitted and order is crucial.

The following are the most typical types of series:

Arithmetic Progression: An arithmetic progression is a set of integers in which any two successive terms' differences are always the same. For instance, any two successive integers in the arithmetic progression sequence 5, 7, 9, 11, 13, and 15 have a common difference of two.

A geometric progression is a series of integers, none of which are zero, where each term after the first is determined by multiplying the preceding term by a predetermined value. The common ratio is the name given to this constant, non-zero value. For instance, the common ratio in the geometric progression 2,6,18,54 is 3, as 2 x 3 = 6, 6 x 3 = 18, and 18 x 3 = 54

An arithmetic progression's reciprocals can be used to create a harmonic progression, which is a different type of progression. One such harmonic progression is the series 1, \(\frac{1}{2}\),\(\frac{1}{3}\), \(\frac{1}{4}\), \(\frac{1}{5}\),\(\frac{1}{6}\)The Fibonacci series is a set of Fibonacci numbers where each number is the sum of the two numbers that came before it. The numbers 0 and 1 begin the series. The sequence goes as: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144…

The formula used for calculating the sum of the first n terms of an arithmetic sequence is as follows:

S_n = \(\frac{n(a_1+a_2)}{2}\)

Where,

n = number of terms

a₁ = the first term

a₂= the last term.