Period of the function Sin z is _____________ degree.

• A) 90
• B) 0
• C) 180
• D) 360

The complex exponential function ez is onto function.

• A) True
• B) False
• C)
• D)

The function ez is an entire function.

• A) False
• B) True
• C)
• D)

Period of the function Cos z is _____________ degree.

• A) 0
• B) 180
• C) 360
• D) 90

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

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

• 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) $$\frac{1}{{{{(1 + Z)}^2}}}$$

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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.