Problem 57 - Calculus

Question

Given that the Taylor series for \(\tan x=x+x^{3} / 3+\) \(21 x^{5} / 120+\cdots,\) then that of \(3 \tan (x / 3)\) is (a) \(3 x+x^{3}+21 x^{5} / 120+\cdots\) (b) \(3 x+x^{3}+21 x^{5} / 40+\cdots\) (c) \(x+x^{3} / 27+7 x^{5} / 3240+\cdots\) (d) \(x+x^{3} / 3+21 x^{5} / 120+\cdots\)

Step-by-Step Solution

Answer

The Taylor series for \(3 \tan(x/3)\) is (c) \(x + x^3/27 + 7x^5/3240 + \cdots\).

Step 1: Substitute and simplify

First, substitute \(x/3\) in place of \(x\) in the Taylor series for \(\tan x\). This gives us the series \(\tan(x/3) = (x/3) + ((x/3)^3)/3 + 21((x/3)^5)/120 + \cdots\).

Step 2: Simplify each term

Next, simplify each term in the substituted series:1. \((x/3)\) is already simplified.2. \(((x/3)^3)/3 = (x^3/27) / 3 = x^3/81\).3. \(21((x/3)^5)/120 = 21(x^5/243)/120 = 21x^5/29160\).

Step 3: Multiply by 3

Since we need the series for \(3\tan(x/3)\), multiply the entire simplified series by 3:- The first term becomes \(3(x/3) = x\).- The second term becomes \(3(x^3/81) = x^3/27\).- The third term becomes \(3(21x^5/29160) = 7x^5/3240\).

Step 4: Reassemble the series

Now, put together the simplified terms into a complete series:\(3\tan(x/3) = x + x^3/27 + 7x^5/3240 + \cdots\).

Key Concepts

Power Series

Understanding a power series is crucial when dealing with functions in calculus. A power series is generally expressed as an infinite sum of terms that have the form \(c_n(x-a)^n\), where \(c_n\) are coefficients, \(x\) is the variable, and \(a\) is the center of the series. Each term is composed of a coefficient and a variable raised to an increasing power. This way, functions can be approximated or represented as sums of simpler polynomial expressions.
Power series are particularly useful because they can represent a wide variety of functions, including polynomials, rational, exponential, and trigonometric functions. In the scope of Taylor and Maclaurin series, the power series becomes a valuable tool because it allows the translation of functions into sums of infinitely many terms centered around a specific point.

Trigonometric Functions

Trigonometric functions like sine, cosine, and tangent relate angles to side ratios in right-angled triangles. These functions are critical in calculus due to their periodic nature, which makes them predictable and mathematically manageable.
The tangent function, \(\tan(x)\), expresses the ratio of the opposite to the adjacent side of a triangle and has a unique Taylor series expansion around zero (Maclaurin series). For example, the Taylor series for \(\tan(x)\) starts as \(x + \frac{x^3}{3} + \frac{21x^5}{120} + \cdots\).
Such expansions allow the manipulation and approximation of trigonometric functions in calculus problems, where exact function values may be difficult to compute.

Calculus Problem Solving

Calculus problem solving often involves finding creative ways to manipulate known series expansions to fit new scenarios. In our Taylor series problem, we begin by adjusting the given series for \(\tan(x)\) to accommodate a scaled function \(3\tan(x/3)\).
This involves substituting the variable and simplifying each term by factoring and reducing fractions. Once the series for \(\tan(x/3)\) is resolved, multiplying it by three allows us to express the target function \(3\tan(x/3)\) accurately.
Each term must be carefully simplified to avoid errors and ensure the integrity of the solution, with particular attention paid to the coefficients and powers.

Mathematical Analysis

Mathematical analysis provides the framework to rigorously explore the properties of functions and their representations. Through analysis, we investigate the convergence of series and the accuracy with which they approximate functions.
With Taylor and Maclaurin series, analysis becomes vital to understanding how well a polynomial approximation can replace a complex function within a specified interval. In our initial function, \(\tan(x)\), analysis ensures that when we substitute \(x/3\) and then scale the result by three, the resulting series will still converge to \(3\tan(x/3)\) for values of \(x\) close to zero.
This steady approach comprises checks on accuracy and convergence, allowing mathematicians to rely on the power of the series in practical applications.