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Calculus · Series-And-Sums

Sum x^n for n = 0 to oo

\sum_{n=0}^{\infty} x^{n}

Stap voor stap

  1. \sum_{n=0}^{\infty} x^{n}

    Write the sum out.

  2. 1 + x + x^{2} + x^{3} + \cdots

    The first few terms.

  3. S_{3} \approx x**2 + x + 1.0, S_{6} \approx x**5 + x**4 + x**3 + x**2 + x + 1.0, S_{11} \approx x**10 + x**9 + x**8 + x**7 + x**6 + x**5 + x**4 + x**3 + x**2 + x + 1.0, S_{51} \approx x**50 + x**49 + x**48 + x**47 + x**46 + x**45 + x**44 + x**43 + x**42 + x**41 + x**40 + x**39 + x**38 + x**37 + x**36 + x**35 + x**34 + x**33 + x**32 + x**31 + x**30 + x**29 + x**28 + x**27 + x**26 + x**25 + x**24 + x**23 + x**22 + x**21 + x**20 + x**19 + x**18 + x**17 + x**16 + x**15 + x**14 + x**13 + x**12 + x**11 + x**10 + x**9 + x**8 + x**7 + x**6 + x**5 + x**4 + x**3 + x**2 + x + 1.0

    Partial sums approach the limit.

  4. = \begin{cases} - \frac{1}{x - 1} & \text{for}\: x > -1 \wedge x < 1 \\\sum_{n=0}^{\infty} x^{n} & \text{otherwise} \end{cases}

    Infinite series: this converges, and the closed form is the limit of the partial sums.

Onthul het antwoord
\begin{cases} - \frac{1}{x - 1} & \text{for}\: x > -1 \wedge x < 1 \\\sum_{n=0}^{\infty} x^{n} & \text{otherwise} \end{cases}