What is the value of $ \sum_{n=1}^{9999}\big [ \dfrac {1} {n^{\frac{1}{4}}}\big ]$ where $[.] $ is the greatest integer function.

Question

What is the value of $ \sum_{n=1}^{9999} \big [ \dfrac {1} {n^{\frac{1}{4}}}\big ]$ where $[\cdot] $ is the greatest integer function.

I do not seem to possess any clue about this problem. Any hints will be deeply appreciated please.

Thank you!

Answer 1

Mh, what about

$$1+0+0+0+0+0+0+0+0+0+0+0+0+0+0+0+0+0+\cdots 0\ ?$$

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If the question is in fact

$$\sum_{1}^{9999}\frac1{\lfloor \sqrt[4]n\rfloor},$$

consider that there are

• $15$ terms (from $1$ to $15$) equal to $1$,

• $65$ terms (from $16$ to $80$) equal to $\dfrac12$,

• $175$ terms (from $81$ to $255$) equal to $\dfrac13$,

• $\cdots$

• $3439$ terms (from $6561$ to $9999$) equal to $\dfrac19$.