QCM : Fundamental Power Series of Mathematical Functions — 6 questions

Questions et réponses du QCM

1. What is the exponential series?

A series expansion of sin x involving alternating signs and factorials
A series used to define hyperbolic functions sinh x and cosh x
A power series expansion of e^x given by ∑_(n=0)^(+∞) (x^n)/(n!)
A geometric series expansion of 1/(1-x) for |x|<1

A power series expansion of e^x given by ∑_(n=0)^(+∞) (x^n)/(n!)

Explication

The exponential series is the power series expansion of e^x, given by the sum of x^n divided by n! from n=0 to infinity, and it converges for all real x.

2. What is the power series expansion of the hyperbolic sine function, sinh(x), as given in the course content?

$ \sinh(x) = \sum_{n=0}^{\infty} \frac{x^{2n+1}}{(2n+1)!}$
$ \sinh(x) = \sum_{n=0}^{\infty} \frac{x^{2n}}{(2n)!}$
$ \sinh(x) = \sum_{n=1}^{\infty} \frac{x^{2n}}{(2n)!}$
$ \sinh(x) = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{(2n+1)!}$

$ \sinh(x) = \sum_{n=0}^{\infty} \frac{x^{2n+1}}{(2n+1)!}$

Explication

The series expansion of hyperbolic sine, sinh(x), is given as $ \sinh(x) = \sum_{n=0}^{\infty} \frac{x^{2n+1}}{(2n+1)!} $, which involves odd powers of x and factorials of odd integers, matching the formula provided in the course content.

3. What is the primary role of the power series expansions of sine and cosine functions?

To define the functions at specific points
To compute derivatives only at zero
To approximate and analyze the functions over their entire domain
To serve as the exact formulas for sine and cosine

To approximate and analyze the functions over their entire domain

Explication

The power series expansions of sine and cosine are primarily used to approximate these functions and analyze their properties over the entire real line, as they converge for all real x and are fundamental in calculus and Fourier analysis.

4. When was the convergence of the exponential series for all real numbers established?

In the early 1700s during Isaac Newton's work
In the late 1800s with the development of rigorous analysis
In the 20th century with modern computational methods
In the 1740s through Leonhard Euler's contributions

In the 1740s through Leonhard Euler's contributions

Explication

Leonhard Euler's work in the 1740s was pivotal in establishing that the exponential series converges for all real numbers, making it a fundamental result in analysis. The other options are incorrect: Newton's work predated the formal proof of convergence, the late 1800s saw further formalizations but not the initial establishment, and the 20th century involved more rigorous analysis rather than the original proof.

5. How are the power series of exponential, hyperbolic, sine, and cosine functions similar in terms of convergence, and how do they differ from the logarithmic series?

They all diverge outside the interval $-1$ to $1$, including the exponential series.
They all converge only within a limited interval, unlike the logarithmic series which converges everywhere.
They all converge for all real numbers, unlike the logarithmic series which converges only within a limited interval.
They all have the same radius of convergence, but the logarithmic series converges only at a single point.

They all converge for all real numbers, unlike the logarithmic series which converges only within a limited interval.

Explication

The exponential, hyperbolic sine and cosine, sine, and cosine series all converge for all real numbers, which is a key similarity. In contrast, the logarithmic series $ ext{ln}(1-x)$ and $ ext{ln}(1+x)$ have a limited interval of convergence, specifically $x ext{ in } ]-1, 1[$, making their convergence domain narrower.

6. Who formulated the geometric series as a sum of powers of x?

Bernhard Riemann
Leonhard Euler
Isaac Newton
Carl Friedrich Gauss

Isaac Newton

Explication

Isaac Newton is credited with formalizing many series expansions, including the geometric series, as part of his work in calculus and infinite series. Euler, Gauss, and Riemann contributed significantly to mathematics but are not primarily credited with the formulation of the geometric series.

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Exponential series — definition?

Series for e^x converging for all real x.

Hyperbolic sine series — expansion?

Sum of x^{2n+1}/(2n+1)! for all real x.

Hyperbolic cosine series — expansion?

Sum of x^{2n}/(2n)! for all real x.

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