Question 1
Find the Taylor expansion of e
ax about x = b.
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![](data:image/png;base64, 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)
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![](data:image/png;base64, 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)
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![](data:image/png;base64, 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)
Question 2
Find the Maclaurin series for ln
![](data:image/png;base64, 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)
. For what values of x does the series converge to the function?
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![](data:image/png;base64, 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)
, for -1 < x < 1
◦
![](data:image/png;base64, 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)
, for -1 < x < 1
◦
![](data:image/png;base64, /9j/4AAQSkZJRgABAQAAAQABAAD/2wBDAAMCAgMCAgMDAwMEAwMEBQgFBQQEBQoHBwYIDAoMDAsKCwsNDhIQDQ4RDgsLEBYQERMUFRUVDA8XGBYUGBIUFRT/2wBDAQMEBAUEBQkFBQkUDQsNFBQUFBQUFBQUFBQUFBQUFBQUFBQUFBQUFBQUFBQUFBQUFBQUFBQUFBQUFBQUFBQUFBT/wAARCAAzAHsDASIAAhEBAxEB/8QAHwAAAQUBAQEBAQEAAAAAAAAAAAECAwQFBgcICQoL/8QAtRAAAgEDAwIEAwUFBAQAAAF9AQIDAAQRBRIhMUEGE1FhByJxFDKBkaEII0KxwRVS0fAkM2JyggkKFhcYGRolJicoKSo0NTY3ODk6Q0RFRkdISUpTVFVWV1hZWmNkZWZnaGlqc3R1dnd4eXqDhIWGh4iJipKTlJWWl5iZmqKjpKWmp6ipqrKztLW2t7i5usLDxMXGx8jJytLT1NXW19jZ2uHi4+Tl5ufo6erx8vP09fb3+Pn6/8QAHwEAAwEBAQEBAQEBAQAAAAAAAAECAwQFBgcICQoL/8QAtREAAgECBAQDBAcFBAQAAQJ3AAECAxEEBSExBhJBUQdhcRMiMoEIFEKRobHBCSMzUvAVYnLRChYkNOEl8RcYGRomJygpKjU2Nzg5OkNERUZHSElKU1RVVldYWVpjZGVmZ2hpanN0dXZ3eHl6goOEhYaHiImKkpOUlZaXmJmaoqOkpaanqKmqsrO0tba3uLm6wsPExcbHyMnK0tPU1dbX2Nna4uPk5ebn6Onq8vP09fb3+Pn6/9oADAMBAAIRAxEAPwD9U6xNS8b+HNG12z0TUNf0ux1m8Cm2065vI47icMSF2RlgzZKsBgckH0rbrzHxz8EIvHWp6+txrlxa6B4jtYbXWdMit4y9wsWdvlzEZiDBircMccoY2+ak79A6HY+FfHfhvxzFcS+HNf0zXo7ZgkzabdxziMnON2wnGQMjPUcjIrdriPBHw4k8Na7ea7qmqjW9bnsYNLW6SzS1VbSF5XiUonBfdM5Zhhem1EGQe3q5JJ6O4v6/r+v8goooqRhRRRQAUV5LCnjLQvjfpVmvjKfxLoup2t1c6nod3ZWsUWkxrtEE1vJFGsoDSfu9kzylwXZSPLbGp4v+N2neCfGcfhjUNB1ttSvrYzaM0EUDRazKCoe2t3Mo2yrvUkT+Uu0s4Yojsoru2mr/AK/r/Mdt/Iv33xn8JaZ46svCN5f3VrrF7L9ntnm026Symn2F/IS8Mf2dptqk+UJN/H3a7evDfFfxH0rxZ8TdJ8NatoHi2207RdSguVk/4Q/VZ4Ly+AzDi6it2gSGJ2DGTzMF0HIRSX8tsF1e38O/8JRp83xVF14f+ISRJaXy67I0uiy3yCUGzlBN3CIWdg5SRkHAZQoUJX5owfXr03ivzd/RX30CouSLl26db2k/ySXq7M+xKxPGXjXQ/h/oF1rfiHUYtM0y2jeWSaQFjtRGkfaqgsxCI7EKCcKT2Neb/G/wz8W9c8U+B5/h/wCJNI0nSLXVll1KG70uadlj+zXKl5mS+gE8O9oh5AXdv2PuwhFXv2mZ7mH9nH4gWgs9Q1jU9Q8P3mnQW2iaXcXks9zNbvGgWGFZHCl2HJyFByzAAmratFO9r/52NIRTqRjLZ/hr/TPULW5ivbWG4hbfDMgkRsEZUjIOD7VLXPfD3Votc8D6HewwX1rHJaRjydSsZrK4QqNpDwzIkiHIPDKMjB6EGuhpzXLJx7GEHzRTCvg7xJ8C/h18Cf2uvP8AF/gTwpr/AMNfitO32W91zRLW5/sTXuphEkkbFYrnkquQPMJ4GCT9414T4oii/aF+Kw8KG2iu/h/4JvYb3Wp5I1kj1DWEKy29kpPG2AFZpSOrNFH0EgrNfHFrf9Ptfht/esW7uEl/V+n+T8m/U6P4Rfs6+Ffgh4y8b6x4PsbPQ9M8UNZyyaLp9otvbWssCOjNGqnaA4ZSVVVAKk87uOj+K3jG58E+DpLuwjWbVry6ttL09HUMoubmdIImYErlVaQOwyMqpAOcVq6Z400XWfFOt+HLK/juNa0WO3l1C1RWzbrOHMO5sbcsI3OASQMEgBhnK+K3g658beDpLSwkWHVrO6ttU093YKpubadJ4lYkNhWaMIxwcKxIGcVTfM1zbafcrbfLYlLlT5fP7/8Ah9yjdfHb4aaH4lXwpqPxJ8KweKElS0bSrvW7SO+aZsbUMO8NvbIwoUZ3DA5FZeieJvEU3x/8VeFL3Vkm0iDw3Y6naRwWqRm3kmu72InJ3FiEgizuO0kEhVBwPTbWV7i2ileF7Z3QM0MpUvGSOVO0kZHTgkehNefW3wjk0r4uar8Rk8WeI768u9PTT30BlsBZGCIyvFEh+zLKCrzSsGM2SXwxKgAOPKtJ9n+X9dy01ytLra33r9LnkUniO8+I/wATPBWm6qfG1u+3WtA1+fw42q22lySQS+XDI8tq3lWzOyNIrl0lUEKWCkBu7+CXjaLwR8DpLv4g63e6JF4f1K+0261jxu8lm3krfSR2jvcXQTzVeJ7cLNlg5bG4tmu2+FPhC78H+F5o9SdW1bUtQu9WvVjcOkU1xM0piRto3KgYICRkha7KpScb33e9tr+Xlv8AeKWsm1trbvb/ADtb7j5e8I/Ez4ReEfFl3rQ/ax03VYby6e7utL1DxD4bME7MCFRpI7ZJ9iDAUCUbQqjpxUvj3x9+z18RtRuNQ1T4++GYtRQwSaRd2ni7TI5NCliOfNszk4Zz/rPN8wOo2MCnyV9OV5NdeM/HfhL4o6fYa6NF1jwrq1rfXMcGiaddDUdMW3VX3yHzJPtaNuWP93FEweSMBW3UOW1+nX08/wAmO7k2+/4hb/tWfBKG3ijf40+Ap3RQplk8T2AZyB947ZAMnrwAPYVJ/wANY/BD/osnw/8A/Cosf/jtXfCn7Q/gTxrrWg6VpWp3z3WvW0l1pj3WjXtrBdqgJkRJpYVjMqBW3RbvMXa25Rg12N74s0rTvE+l+Hbi68vWNUt7i6tLby3PmxQGIStuA2jaZouCQTu4zg4pRctl/S3+4lLojgf+Gsfgh/0WT4f/APhUWP8A8do/4ax+CH/RZPh//wCFRY//AB2tH44ePtb+G3hO21nSLSwuY/7S0+yuDes5IW5vre2+RFxk7ZnbJYYKL8rZOPRKVnbm6f8ADf5jaaVzlPAvxY8D/FD7d/whvjLw/wCLfsOz7X/YWqQXv2ffu2eZ5Ttt3bHxnGdrY6GuroopCCvmDWP+CZ/7N3iDV77VNR+Hb3moX073NzcS+INTLyyuxZ3J+09SST+Nafj34LWPxC+OPi1ta+FcGreHNV8Kx2T63epYSW15exyO8ZaMzecXRWVUlaPKkMFZRy2Bp37PPxH8b+CfBFzeeK7vwPqlj4fstNv9Jv77WpJhcwR+XJIz6Zr1vbtvI3BghYgjc2flXOD51zNWadvld/5J/NFSfLLkT0/W0X/7c18manhPw18JP2Kn8SaZ8Pfh9rkdmbaHV/Es2jXD36adbjzVinmF1c+YeEmOy3WR8KSUxgn6QsryHUbOC7tpBLbzxrLHIOjKwyD+INfOWpeAPiF4Ls/FHh+y0KTxxF4n8O2+kprltf8AlLZ3SRTQvJdC+u5LgxESowZJLiT5XBHQn6F0DSV0DQdN0xJGmSyto7ZZH6sEUKCfc4rRSlK/N8n+FvRJK3r5GaXKo2Vr3uu21vvu7+n336KKKCgooooAK8k8O/Dv4m6P4u1bWbzxx4S1Jb8yDLeEblLmKLDfZ4Vl/tMqI42IJURjf8xJDOXHrdFBSbSseI+FPgF4j8PaD8LLS58YaZfah4I1Ka7e6TQpIo7+3lgngaIR/a2MUmydiJNzrkA+WRxW94n/AGZvhp4w+J+leP8AVvB+i3viOwjlXz5tMtpPtDsYSk0zNGXeSLyQI23ApvfHXj1CiiPuW5en+ViLb+Z5x8d/hrr/AMV/BtvoOheItN8NkajZ381xqOkSahv+zXEdxGiqlzBtzJCm4ktldwABIYd/YJdR2Nsl7NDcXqxqJ5reIxRvJgbmVCzFVJyQpZiBxk9asUU7u3L03KbvYKKKKQgooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKAP/Z)
, for -1 < x < 1
◦
![](data:image/png;base64, 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)
, for -1 < x < 1
◦
![](data:image/png;base64, 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)
, for all x