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Is this Convergent?

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

Note by Danny Kills
2 years, 11 months ago

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

There is no exact expression for this constant. · 2 years, 11 months ago

Sir, how do we find out whether a sum like this converges or not? Thanks. · 2 years, 11 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. · 2 years, 11 months ago

Comment deleted Aug 28, 2015

Is it a typo where $$r$$ should be $$i$$? But even so, the terms approach $$1$$ for large $$i$$, so this doesn't converge. · 2 years, 3 months ago