Is this Convergent?

Does this sum converge? If yes, then find its value.

#Algebra #InfiniteSum #NestedRadicals #RamanujanSummation

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Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

This is the "nested radical constant", also known as Vijayaraghavan's constant, which is $1.75793275...$

There is no exact expression for this constant.

Michael Mendrin - 6 years, 10 months ago

Sir, how do we find out whether a sum like this converges or not? Thanks.

Satvik Golechha - 6 years, 10 months ago

Look up "Herschfeld's Convergence Theorem". It states that for real terms ${ { x }_{ n } }\ge 0$ and real powers $1>p>0$ if and only if

$\displaystyle{ { x }_{ n } }^{ { p }^{ n } }$

is bounded, then the following converges

${ x }_{ 0 }+{ \left( { x }_{ 1 }+{ \left( { x }_{ 2 }+{ \left( ...+{ \left( { x }_{ n } \right) }^{ p } \right) }^{ p } \right) }^{ p } \right) }^{ p }$

as $n\rightarrow \infty$

Here, $p=\frac { 1 }{ 2 }$ , so it's easy to see how the condition is met.

Michael Mendrin - 6 years, 10 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$