Expanding 23x?2 in Ascending Powers of 'x' and Determining the Range of Validity

When it comes to expanding expressions involving negative exponents in ascending powers of a variable lsquo;xrsquo;, the binomial expansion provides a powerful method. We will explore how to expand 23x?2 in ascending powers of 'x' up to the term including x2 and determine the range of validity for this expansion.

Conceptual Understanding

The binomial series expansion formula for (1 u)? provides a systematic way to express such expansions. However, for u x?2, the formula needs to be adjusted. Typically, the binomial series is given as:

1 u? 1 nu n!(n-1)!2!u2 ...

For expressions where the exponent is negative, such as 23x?2, the expansion can be rewritten using this concept.

Step-by-Step Expansion Using the Binomial Series

Let's begin by rewriting the given expression, 23x?2, in a form suitable for the binomial series expansion.

23x?2 0.25(1 3x2)-2

Applying the binomial series expansion to (1 u)?2 where u 3x2, we have:

(1 u)?2 1 2u (-2)!2!(-3)!3!u2 ...

Substituting u 3x2:

(1 3x2)-2 1 2( 3x2 ) (-2!)(-3!3!)2! (-9x24) ...

Simplifying each term:

Constant term: 1 First-order term: 2( 3x2 ) 3x Second-order term: (-2!)(-3!3!)2! ( -9x24 ) (-4)(24/6) ( 9x216 ) -4824 ( 9x216 ) -27x216

Multiplying the entire series by 0.25:

23x?2 0.25(1 3x -27x216) 0.25 0.75x 0.421875x2

Range of Validity

The binomial expansion is valid for . In this case, u 3x2.

Therefore, the expansion is valid when:

3x2 1

Which simplifies to:

3x 2

Hence:

x 23

This implies that the range of x for which the expansion is valid is:

-23 x 23

Conclusion

By using the binomial series expansion, we have successfully expanded 23x?2 in ascending powers of 'x' up to the term including x2. The range of x for which this expansion is valid has been determined to be -23 x 23. This method provides a systematic way to handle such expansions in calculus and related fields.