Chapter 2: Q31MP (page 81)

Show that if a sequence of complex numbers tends to zero, then the sequence of absolute values tends to zero too, and vice versa. Hint $a_{n} + {ib}_{n} \to 0$ means $a_{n} \to 0$ and $b_{n} \to 0 .$

Short Answer

Expert verified

It has been proved that $a_{n} + {ib}_{n} \to 0$ means

Step by step solution

Given Information.

It has been given that the complex numbers tends to zero.

Meaning of rectangular form.

Represent the complex number in rectangular form means writing the given complex number in the form of $x + iy$ in which x is the real part and y is the imaginary part.

Solve the series.

Assume a sequence of complex number.

$S = \sum C_{n} S = C_{1} + C_{2} + C_{3} + \dots$

A complex number has a real and an imaginary part.

$S = \sum C_{n} S = \sum a_{n} + i b_{n} S = \sum a_{n} + i \sum b_{n}$

If tends to be zero thentends to be zero and this is only possible when must be zero.

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Show that if a sequence of complex numbers tends to zero, then the sequence of absolute values tends to zero too, and vice versa. Hint $a_{n} + {ib}_{n} \to 0$ means $a_{n} \to 0$ and $b_{n} \to 0 .$

Short Answer

Step by step solution

Given Information.

Meaning of rectangular form.

Solve the series.

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Show that if a sequence of complex numbers tends to zero, then the sequence of absolute values tends to zero too, and vice versa. Hintan+ibn→0means an→0andbn→0.

Short Answer

Step by step solution

Given Information.

Meaning of rectangular form.

Solve the series.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Rotational Dynamics

Classical Mechanics

Thermodynamics

Fluids

Energy Physics

Electric Charge Field and Potential

Study anywhere. Anytime. Across all devices.

Show that if a sequence of complex numbers tends to zero, then the sequence of absolute values tends to zero too, and vice versa. Hint $a_{n} + {ib}_{n} \to 0$ means $a_{n} \to 0$ and $b_{n} \to 0 .$