21.60 A function f(x)=3* +3. 3* +48• 3* + . 3* +..., defined in the set R, is given, and the right side of the279√5formula of this function is a convergent geometric series. The function takes a valuethe interval (k, k+1), where kEZ.a) Find k.for some argument belonging

Answered: 4 1.60 A function f(x)=3* + • 3* + . 3*…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

2
1.60 A function f(x)=3* +
3
. 3* +
4
8
• 3* + . 3* +..., defined in the set R, is given, and the right side of the
27
9
√5
formula of this function is a convergent geometric series. The function takes a value
the interval (k, k+1), where kEZ.
a) Find k.
for some argument belonging

Expert Solution

Step 1

What is Infinite Geometric Series:

An infinite number of terms with a fixed ratio between them are added to form a geometric series. Geometric series can be used as an introduction to frequently used mathematical tools including the Taylor series, the complex Fourier series, and the matrix exponential. It also plays a significant part in the early development of calculus and is utilised across mathematics.

Given:

Given function is $f (x) = 3^{x} + \frac{2}{3} \cdot 3^{x} + \frac{4}{9} \cdot 3^{x} + \frac{8}{27} \cdot 3^{x} + . . . . . . . .$ . It is also given that the function takes a value $\sqrt{5}$ for some argument belonging the interval $(k, k + 1)$ .