View Attachment to Understanding Taxes

[anil86]

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[chisigma]

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Wellcome on MHB anil86!... You can start from the binomial sum...

$\displaystyle \frac{1}{\sqrt{1 - x^{2}}} = 1 + \frac{1}{2}\ x^{2} + \frac{3}{8}\ x^{4} + ...\ (1)$

... and from (1) ...

$\displaystyle \frac{x}{\sqrt{1 - x^{2}}} = x + \frac{1}{2}\ x^{3} + \frac{3}{8}\ x^{5} + ... \ (2)$

... so that the sum of Your series is... $\displaystyle S = \text{Im} \{\frac{e^{i\ \theta}}{\sqrt{1 - e^{2\ i\ \theta}}}\}\ (3)$

Are You able to proceed?...

Kind regards

$\chi$ $\sigma$

 

[anil86]

Wellcome on MHB anil86!... You can start from the binomial sum...

$\displaystyle \frac{1}{\sqrt{1 - x^{2}}} = 1 + \frac{1}{2}\ x^{2} + \frac{3}{8}\ x^{4} + ...\ (1)$

... and from (1) ...

$\displaystyle \frac{x}{\sqrt{1 - x^{2}}} = x + \frac{1}{2}\ x^{3} + \frac{3}{8}\ x^{5} + ... \ (2)$

... so that the sum of Your series is... $\displaystyle S = \text{Im} \{\frac{e^{i\ \theta}}{\sqrt{1 - e^{2\ i\ \theta}}}\}\ (3)$

Are You able to proceed?...

Kind regards

$\chi$ $\sigma$

Please view attachment!View attachment 1687

 

[chisigma]

Wellcome on MHB anil86!... You can start from the binomial sum...

$\displaystyle \frac{1}{\sqrt{1 - x^{2}}} = 1 + \frac{1}{2}\ x^{2} + \frac{3}{8}\ x^{4} + ...\ (1)$

... and from (1) ...

$\displaystyle \frac{x}{\sqrt{1 - x^{2}}} = x + \frac{1}{2}\ x^{3} + \frac{3}{8}\ x^{5} + ... \ (2)$

... so that the sum of Your series is... $\displaystyle S = \text{Im} \{\frac{e^{i\ \theta}}{\sqrt{1 - e^{2\ i\ \theta}}}\}\ (3)$

Is...

$\displaystyle 1 - e^{2\ z} = - e^{z}\ (e^{z} - e^{- z}) -> \sqrt{1 - e^{2\ z}} = i\ e^{\frac{z}{2}}\ \sqrt {e^{z} - e^{- z}}\ (1)$

... so that for $\displaystyle z = i\ \theta $ is...

$\displaystyle \text{Im}\ \{\frac{e^{i\ \theta}}{\sqrt{1 - e^{2\ i\ \theta}}}\} = \text{Im}\ \{ \frac{e^{i\ \frac{\theta}{2}}}{i\ \sqrt{e^{i\ \theta} + e^{- i\ \theta}}} \} = \frac{\sin \frac{\theta}{2} + \cos {\frac{\theta}{2}}}{2\ \sqrt{\sin \theta}}\ (2)$

Kind regards

$\chi$ $\sigma$

 

[CellCoree]

Thank you for sharing this guide on understanding taxes. I appreciate the clear and concise information presented in this document. Taxes play a crucial role in funding important government programs and services, and it is important for individuals to have a basic understanding of how they work.

I particularly appreciate the section on the different types of taxes, including income, sales, and property taxes. This provides a comprehensive overview of the various ways in which individuals and businesses contribute to the government's revenue.

In addition, the guide does a great job of explaining the different tax brackets and how they impact an individual's tax liability. This is important information for individuals to understand as they plan their finances.

Overall, this guide serves as a valuable resource for individuals looking to gain a better understanding of taxes. I would recommend it to anyone looking to educate themselves on this important aspect of our society.