diff --git a/.obsidian/plugins/obsidian-git/data.json b/.obsidian/plugins/obsidian-git/data.json index 4bc189e..e69de29 100644 --- a/.obsidian/plugins/obsidian-git/data.json +++ b/.obsidian/plugins/obsidian-git/data.json @@ -1,27 +0,0 @@ -{ - "commitMessage": "vault backup: {{date}}", - "autoCommitMessage": "vault backup: {{date}}", - "commitDateFormat": "YYYY-MM-DD HH:mm:ss", - "autoSaveInterval": 5, - "autoPushInterval": 0, - "autoPullInterval": 5, - "autoPullOnBoot": false, - "disablePush": false, - "pullBeforePush": true, - "disablePopups": false, - "listChangedFilesInMessageBody": false, - "showStatusBar": true, - "updateSubmodules": false, - "syncMethod": "merge", - "customMessageOnAutoBackup": false, - "autoBackupAfterFileChange": false, - "treeStructure": false, - "refreshSourceControl": true, - "basePath": "", - "differentIntervalCommitAndPush": false, - "changedFilesInStatusBar": false, - "showedMobileNotice": true, - "refreshSourceControlTimer": 7000, - "showBranchStatusBar": true, - "setLastSaveToLastCommit": false -} \ No newline at end of file diff --git a/education/math/MATH1220 (calc II)/Sequences.md b/education/math/MATH1220 (calc II)/Sequences.md index 576f129..e1bc0d3 100644 --- a/education/math/MATH1220 (calc II)/Sequences.md +++ b/education/math/MATH1220 (calc II)/Sequences.md @@ -111,5 +111,10 @@ Then if the *series converges* absolutely then the sum converges. > Does the series $\sum_{n=1}^\infty (\frac{(-1)^n}{n+5}$ conditionally converge, absolutely converge, or diverge? 1. Consider $\sum_{n=1}^\infty|\frac{(-1)^n}{n+5}| = \sum_{n=1}^\infty \frac{1}{n+5}$. -2. The above series can be compared to $\frac{1}{n}$ with the limit comparison test. +2. The above series can be compared to $\frac{1}{n}$ with the limit comparison test. $\lim_{n \to \infty}\dfrac{\frac{1}{n}}{\frac{1}{n+5}} = 1 \ne 0, \infty$. By the L.C.T, $\frac{1}{n+5}$ diverges, so the series does not absolutely converge. +3. Applying the Alternating Series Test, + - $a_n > 0$ for all $n$ - all values in the series are greater than zero + - $a_n \ge a_{n+1}$ - the series decreases + - $\lim_{n \to \infty} a_n = 0$ - the series approaches zero +All three conditions hold true, therefore we know that $\sum_{n=1}^\infty \frac{(-1)^n}{n+5}$ conditionally converges.