# Thread:

1. ## Taylor and Maclaurin series

Find the Taylor and Maclaurin series of the given function with the given point $z_0$ as center and determine the radius of convergence.

$e^{\frac{z^2}{2}}\int e^{-t^2} dt$

By the way, integrate from 0 to z and center at $z_0 = 0$

I cannot imagine, how can I solve this.

2. Originally Posted by noppawit
Find the Taylor and Maclaurin series of the given function with the given point $z_0$ as center and determine the radius of convergence.

$e^{\frac{z^2}{2}}\int e^{-t^2} dt$

By the way, integrate from 0 to z and center at $z_0 = 0$

I cannot imagine, how can I solve this.
centered at 0 ...

$e^u = 1 + u + \frac{u^2}{2!} + \frac{u^3}{3!} + ...$

$e^{-t^2} = 1 - t^2 + \frac{t^4}{2!} - \frac{t^6}{3!} + ...$

$\int_0^z e^{-t^2} \, dt = \left[t - \frac{t^3}{3} + \frac{t^5}{5 \cdot 2!} - \frac{t^7}{7 \cdot 3!} + ... \right]_0^z = z - \frac{z^3}{3} + \frac{z^5}{5 \cdot 2!} - \frac{z^7}{7 \cdot 3!} + ...$
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 9, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.8938313722610474, "perplexity_flag": "middle"}