$${\text{For the power series }}\sum\limits_{n = 0}^\infty {{Z^n}} {\text{ is equal to __________}{\text{.}}$$

• A) 1/(1-Z)
• B) 1/(1+Z)
• C) Z
• D) 1/Z

The complex exponential function ez is one-to-one function.

• A) True
• B) False
• C)
• D)

In the context of an infinite power series, what ensures the existence of the corresponding function?

• A) The series having an infinite number of terms.
• B) The series having only positive coefficients.
• C) The divergence of the series.
• D) The convergence of the series.

$${\text{The power series }}\sum\limits_{n = 0}^\infty {{C_n}{{(Z - \alpha )}^n}} {\text{converges for all values of z for which |Z - }}\alpha {\text{| _________}}{\text{.}}$$

• A) less than 1
• B) less than equal to 1
• C) less than radius of convergence
• D) less than equal to radius of convergence

$${\text{For the power series }}\sum\limits_{n = 0}^\infty {(n + 1){Z^n}} {\text{ is equal to _________}}{\text{.}}$$

• A) $$\frac{1}{{{Z^2}}}$$
• B) $$\frac{1}{Z}$$
• C) $$\frac{1}{{{{(1 - Z)}^2}}}$$
• D) $$\frac{1}{{{{(1 + Z)}^2}}}$$

data:image/png;base64,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 .

• A) log z
• B) ez
• C) cos z
• D) sin z

Which one of the following is the Euler’s formula?

• A) data:image/png;base64,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 .
• B) data:image/png;base64,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 .
• C)
• D)

The ratio test for the power series $\sum_{n=0}^{\infty} a_n x^n$ involves computing which of the following limits?

• A) $\lim_{n \to \infty} \left| \frac{a_n x^n}{a_{n+1} x^{n+1}} \right|$
• B) $\lim_{n \to \infty} \left| \frac{a_n}{a_{n+1}} \right|$
• C) $\lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|$
• D) $\lim_{n \to \infty} \left| a_n x^n \right|$

$${\text{The power series for }}{C_n} = n!{\text{ }}and{\text{ }}\alpha = 5{\text{ is _____________}}{\text{.}}$$

• A) $$\sum\limits_{n = 0}^\infty {5!{{(Z - 5)}^n}}$$
• B) $$\sum\limits_{n = 0}^\infty {{{(Z - 5)}^n}}$$
• C) $$\sum\limits_{n = 0}^\infty {{Z^n}}$$
• D) $$\sum\limits_{n = 0}^\infty {n!{{(Z - 5)}^n}}$$

$${\text{In an infinite series of the form }}\sum\limits_{n = 0}^\infty {{C_n}{{(Z - \alpha )}^n}} {\text{ __________ is not a fixed complex number}}{\text{.}}$$

• A) $${\alpha ^n}$$
• B) $${C_n}$$
• C) $$\alpha$$
• D) $$Z$$

The complex exponential function w = ez is one-to-one if we only consider ___________.

• A) the principal argument
• B) the argument
• C)
• D)

$${\text{The radius of convergence of power series }}\sum\limits_{n = 0}^\infty {{{(\frac{{n + 15}}{{6n + 10}})}^n}{{(Z - 5)}^n}} {\text{ is __________}}{\text{.}}$$

• A) 15
• B) 5
• C) 10
• D) 6

$${\text{The infinite series }}\sum\limits_{n = 0}^\infty {{C_n}{{(Z - \alpha )}^n}} {\text{ defines a function only if it _________}}{\text{.}}$$

• A) diverges
• B) converges
• C)
• D)

How many methods are there to find the radius of convergence of power series?

• A) 1
• B) 4
• C) 2
• D) 3

Which of the following correctly describes the radius of convergence $R$ for the power series $\sum_{n=0}^{\infty} a_n x^n$?

• A) $R$ is the distance from the origin to the nearest point where the series diverges.
• B) $R$ is the maximum value of $x$ for which the series converges absolutely.
• C) $R$ is the maximum value of $|x|$ for which the series converges absolutely.
• D) $R$ is the distance from the origin to the nearest point where the series converges.

Why is the infinite series used in constructing complex valued functions?

• A) To simplify the calculations
• B) To make them shorter
• C) To improve convergence
• D) To make them infinitely long

Period of the function ez is _____________ degree.

• A) 360
• B) 180
• C) 0
• D) 90

Under what condition does an infinite power series define a function?

• A) When the series has an odd number of terms.
• B) When the series is convergent.
• C) When the series is divergent.
• D) When the series has a constant term.

$${\text{For any complex number z, }}\frac{d}{{dz}}{\text{(csc z) = ____________}}{\text{.}}$$

• A) $$- \cot z\csc z$$
• B) $$\cot z\operatorname{cscz}$$
• C) $$\tan z\sec z$$
• D) $$\cot z\sec z$$

$${\text{For any complex number z, }}\frac{d}{{dz}}{\text{sinh }}z{\text{ = ___________}}{\text{.}}$$

• A) $$\tanh z$$
• B) $$\sec {h^2}z$$
• C) $$\cosh z$$
• D) $${\csc ^2}z$$